Astrophysics

Of course, when we actually do so, the values we obtain dwarf anything we have direct experience with, thus stretching our imagination, and challenging the physical intuition and insights we instinctively draw upon to function in our own everyday world.

And since the Greeks decided to give dimmer stars a higher magnitude, we find that magnitude scales with the log of the inverse flux.

As a mneumonic, this is cast as a 10-digit "telephone number", with the 3-digit "area code" representing the 3 steps of 10-5 from us down to the nucleus, and 7-digit main-number representing 7 key steps to the scale of the universe.

The full sequence of steps over this span thus looks something like a 10-digit phone number with area code: 555-711-2555.

The apparent angular size that object subtends in our overall field of view is then used intuitively by our brains to infer the object's distance, based on our extensive experience that a greater distance makes the object subtend a smaller angle.

Neurosensors in the eye muscles that effect this inward pointing relay this inward angle to our brain, where it is processed to provide our sense of 'depth' perception.

The key point here is that the parallax angle shift of your finger, which results from switching perspective from one eye to the other, exactly fits the apparent angular separation between your own mirror-image eyes.

The parallax for even the nearest star is less than an arcsec, implying stars are all at distances more (generally much more) than a parsec.

Integration over a full sphere shows that there are 4π steradians in the full sky.

This is a profoundly important equation in astronomy, and so you should not just memorize it, but embed it completely and deeply into your psyche.

Thus we see that the Sun emits the power of about 4 x 10^24 100-watt light bulbs! In common language this corresponds to four million billion billion, a number so huge that it loses any meaning.

The huge flux from this large, bright solid angle would cause a lot more than a mere sunburn!

The light they emit is called "thermal radiation", and arises from the jostling of the atoms (and particularly the electrons in and around those atoms) by the violent collisions associated with the star's high temperature.

Each photon carries a discrete, indivisible 'quantum' of energy that depends on the wave frequency.

This is not entirely coincidental, since our eyes evolved to use the wavelengths of light for which the solar illumination is brightest.

The Stefan-Boltzmann law is one of the linchpins of stellar astronomy.

As such the associated wavelengths of the absorption lines in a star's spectrum provide a direct 'fingerprint' — perhaps even more akin to a supermarket bar code — for the presence of that element in the star's atmosphere.

The associated wavelengths of the absorption lines in a star's spectrum provide a direct 'fingerprint'—perhaps even more akin to a supermarket bar code—for the presence of that element in the star's atmosphere.

In summary, the appearance of absorption lines in stellar spectra provides a real treasure trove of clues to the physical properties of stars.

The reason there are so many stars in this main-sequence band is that it represents the long-lived phase when stars are stably burning Hydrogen into Helium in their cores.

Imagine what you'd weigh then on the surface of a neutron star!

Imagine what you'd weigh then on the surface of a neutron star!

This fact that the total energy E just equals half the gravitational binding energy U is an example of what is known as the Virial Theorem.

Even in the 19th century, it was clear, e.g. from geological processes like erosion, that the Earth — and so presumably also the Sun — had to be much older than this.

The most massive stars, of order 100 M, and thus with luminosities of order 106L, have main-sequence lifetimes of only about about 1 Myr, much shorter the multi-Gyr timescale for solar-mass stars.

This rejuvenated the mass gainer, making it again a hot, luminous blue star.

Consider the noise from a car on a highway, for which the 'vvvvrrrrrooomm' sound stems from just this shift in pitch from the car engine noise.

It turns out, in fact, that stellar binary (and even triple and quadruple) systems are quite common, so much so that astronomers sometimes joke that 'three out of every two stars is (in) a binary'.

In visual binaries, monitoring of the stellar positions over years and even decades reveals that the two stars are actually moving around each other, much as the Earth moves around the Sun.

However, if the orbital plane is not perpendicular to the line of sight, then the orbital velocities of the stars will give a variable Doppler shift to each star's spectral lines.

In eclipse jargon, the times when the stellar rims just touch are called 'contacts', labeled 1-4 for first, second, etc.

Indeed, to get good results, one often has to relax even the assumption that the stars are spheres with uniform brightness, taking into account the mutual tidal distortion of the stars.

In hotter, more-massive stars, the rotation can be more rapid, typically 100 km/s or more, with some cases (e.g., the Be stars) near the "critical" rotation speed at which material near the equatorial surface would be in a Keplerian orbit!

When Galileo first used a telescope to magnify the apparent disk of the Sun, he found it was not the 'perfect orb' idealized from antiquity, but instead had groups of relatively dark 'sunspots' spread around the disk.

As the solid angle of the projected emitting area declines with the inverse square of the distance, the fixed solid angle receiving the intensity grows in area in proportion to the distance-squared.

It is clear from the initial definition that one can think of optical depth as simply the number of mean-free-paths between two locations.

Because this redder, longer wavelength light is less strongly absorbed than the bluer, shorter wavelengths, the remaining light tends to appear " reddened ", much in the same way as the Sun's light at sunset.

The scattering of blue light out of the direction from the Sun makes the sunset red, while all that scattered blue light makes the sky blue.

But this can be reduced to resolutions approaching 0.1 arcsec through a technique called adaptive optics, wherein reflection from a laser beam shot up into the sky is used to estimate these seeing distortions, and then dynamically deform secondary mirrors to correct for them.

An extension of this technique, called Very Long Baseline Interferometry (VLBI) can even combine signals from telescopes spread all around the globe; their diffraction limit can thus in principle approach that of a telescope the size of the entire Earth!

These vividly demonstrate that the Sun is in fact highly structured and variable over a wide range of spatial and temporal scales, and so provide a sobering reality check on our own simple idealizations of stars as being constant, featureless, spherically symmetric balls of gas.

This is followed by an abrupt jump across a narrow transition region to temperatures of millions of Kelvin (!) in the solar corona.

The lack of a strong field on Mars has allowed the solar wind to gradually erode its now much thinner atmosphere.

Such a conducting plasma makes any magnetic field 'frozen-in', or effectively stuck to the local plasma.

The latter refers to the fact that the Sun does not rotate as a solid body, like the Earth or any planet, but instead actually has a faster angular rotation at its equator than at higher latitudes towards its poles.

For the solar atmosphere, the sound speed is cs ≈ 9 km/s, about 1/60th of the surface escape speed Vesc = 620 km/s.

Thus, while surface temperatures of stars are typically a few thousand Kelvin, we see that their interior temperatures are typically of order 10 million Kelvin!

The Sun wouldn't quite float in your bathtub.

Photons created in the core of the Sun need to scatter a total of N ≈ 2 1022 times to reach the surface!

The terms in square bracket can be thought of as a radiative conductivity, which we note increases with the cube of the temperature T³, but depends inversely on opacity and density.

Two remarkable aspects of this derivation are that: (1) the role of the stellar radius cancels; and (2) the resulting M-L scaling does not depend on the details of the nuclear generation of the luminosity in the stellar core!

Convection refers to the overturning motions of the gas, much like the bubbling of boiling water on a stove.

In the dense and hot stellar interior, once convection sets in, it is so efficient at transporting energy that it keeps the local temperature gradient very close to the adiabatic value above which it is triggered.

The higher charge of CNO nuclei requires higher proton energy to overcome the higher electrical repulsion.

Within quantum mechanics, it thus has an associated 'fuzziness' in position, characterized by its De Broglie wavelength λ = h/p, where h is Planck's constant.

Stars with a mass below this minimum should not be able to ignite H-fusion, because electron degeneracy prevents their cores from contracting to a small enough size to reach the 17 MK temperature required for fusion.

This gives the radiative acceleration the same inverse-square radial decline as the stellar gravity, g=GM=r2, meaning that it acts as a kind of \anti-gravity".

Then, much as the hot coals at the heart of a wood fire help burn the wood fuel around it much faster, the higher temperature of a contracted stellar core actually makes the overlying shell of Hydrogen fuel around the core burn even more vigorously!

The star's luminosity thus increases, with now the temperature staying nearly constant at the cool value for the Hayashi limit.

In contrast, for a degenerate gas, the expansion from adding heat actually makes the temperature increase even further.

This small radius makes them very dense, with a density about a million times (!) the density of water, and so a million times the density of normal main-sequence stars like the Sun.

But note that this radius actually decreases with increasing mass.

This eventually leads to a catastrophic core collapse, halted only when the electrons merge with the protons in the Iron nuclei to make the entire core into a collection of neutrons.

For a few weeks, the luminosity of such a supernova can equal or exceed that of a whole galaxy, up to 10^12 L_sun!

And much as a sufficiently dense, heavy ball could rip a hole in the trampoline, for objects with mass concentrated within a radius Rbh, the bending becomes so extreme that it effectively punctures a hole in space-time.

By studying the resulting high-energy radiation, we can infer the presence and basic properties (mass, even rotation rate) of black holes in such binary systems, even though we can't see the black hole itself.

Thus, for example, over a length of 1 km, this would require measurement of distance changes on a scale of 10-18m, or about 1/1000 the size of a proton!

The final, merged black hole was inferred to be about 62 M, with the extra 3M converted to energy in the emitted gravitational wave, which for brief, few msec of the merger represented some 50 times the luminosity of all the stars in the observable universe!

This supported earlier suggestions that such high-mass elements, colloquially characterized as 'bling' because of their prominent use as precious metals and in jewelry, are mostly produced in the 'kilonovae' associated with such neutron-star mergers.

Thus while most stars typically have mean densities comparable to matter (like water) here on Earth (i.e., ρ ≈ 1 g/cm3), this ISM density is well below (by a factor ∼ 10^4) even the most perfect vacuum ever created in terrestrial laboratories.

In the absence of any restraining force, and ignoring any disturbances from stellar mass ejection, the ISM gas should over time settle into a dynamical equilibrium that is roughly isobaric.

In such SN explosions, several solar masses of stellar material is ejected at very high speeds, approaching 10% the speed of light.

The dust itself is generally not formed locally, since in even the coldest clouds the density is not high enough for efficient nucleation of microscopic dust grains.

Stars are typically formed out of interstellar gas and dust in very dense molecular clouds, which often have 10 or 20 magnitudes of visual extinction, essentially completely obscuring them at visual wavelengths.

Viewed in the visible, HII regions thus generally have a distinctly reddish glow, as illustrated in the left panel of

Viewed in the visible, HII regions thus generally have a distinctly reddish glow, as illustrated in the left panel of figure 21.4 for the HII region known as the Rosetta nebula.

But a general upshot of such a large Jeans mass is that stars tend typically to be formed in large clusters, resulting from an initial contraction of a GMC, with mass of order 10^4 M_sun or more.

But in practice CO emission is often so efficient that the cloud interior can stay cool, or even become cooler, as it contracts.

The large power-index reflects the fact that higher-mass stars are much rarer than lower-mass stars.

You and I and everyone on Earth are here today because our source material happened to stem from the equatorial regions of the proto-solar core, with too much angular momentum to fall into the Sun itself.

and everyone on Earth are here today because our source material happened to stem from the equatorial regions of the proto-solar core, with too much angular momentum to fall into the Sun itself

The disk gaps likely represent regions of planet formation.

The disk gaps likely represent regions where planet formation is clearing out disk debris, though there is so far no direct evidence of fully formed planets in this system.

In the colder outer regions these condensed to form ice, which gradually collected into ever larger solid cores, eventually growing massive enough to gravitationally attract and retain the even-more-abundant but lighter gases of Hydrogen and Helium.

But that is the result of a somewhat fortuitous and delicate cancellation, between the cooling effect of reflection of sunlight by clouds, and the warming effect of greenhouse gases in the Earth's atmosphere.

This sent the icy minor bodies hurtling toward the inner solar system, to impact the moon, and of course also the Earth.

Instead it just guides some fraction of solar particles to impact near the magnetic poles, where they harmlessly light up the upper atmosphere to form the beautiful dance of the northern and southern lights.

This greatly complicates direct detection of extra-solar planets, since this reflected light is generally overwhelmed by the direct light from the star.

In the context of the prevailing idea that such gas giants should only form beyond the ice line, this detection was a real surprise.

For an Earth-size planet around a solar-type star, the alignment must be within an angle ~0.27 degrees.

These range from the intermediate size Ice Giants, which when closer to their star become "Ocean Worlds", to Rocky Planets, which when very close can become "Lava Worlds", as the rocks are melted by intense heating from the star's radiation.

In particular, any signature of molecular oxygen would be viewed as an indicator for life, since this is normally very reactive and would be destroyed unless constantly being replenished by photosynthesis.

We along with our Sun are today still embedded within the Milky Way's disk, orbiting about the galactic center, again because our bits of proto-galactic matter had too much angular momentum to fall further inward.

As we look up into a dark night sky, we can trace clearly the direction along this disk plane through the faint milky glow of thousands of distant, unresolved stars, from which we indeed get the name 'Milky Way'.

They are so loosely bound that they tend to disperse within a few 10 Myr or less, evolving into unbound OB associations.

The results indicate that, inside the Sun's orbit (R < Ro), the rotation speed is nearly constant, with V(R) ≈ Vo.

This comparison illustrates why these flat rotation curves came as a surprise.

In short, without dark matter, we wouldn't be here today to wonder about it!

The star S0-2 has the shortest period, 16.7 years, and so has been tracked over more than a full orbit.

In particular, he was able to measure the Doppler shift of known spectral lines, giving then a direct measure of the galaxies' radial velocity Vr.

Einstein's limit really applies to how fast objects can travel relative to space, but that space itself can expand at a speed faster than light!

Nonetheless, as a strictly empirically calibrated relation, this Tully-Fisher scaling provides a luminous standard candle to infer distances beyond the range accessible to the Cepheid method.

For individual stars, the distance/size ratio is enormous, of order d/s ~ 4 x 10^7, implying a mean-free-path l ~ 10^15 pc!

When galaxies do collide, their overall pattern of stars become strongly distorted by the mutual tidal interaction of the overall mass of the two galaxies; but the individual stars are too widely separate to collide, and so just pass by each other.

This extreme luminosity from such a small volume is thought to be the result of matter accreting onto the supermassive black hole (SMBH) at the center of the QSO/AGN host galaxy.

In essence, the huge luminosities and huge distances of quasars provide us a set of "flashlights" to probe the inter-galactic Hydrogen gas in the universe between us and the quasars.

Quite remarkably, individual clumps in these quasar jets can sometimes show an apparent 'super-luminal' motion, meaning that, for the inferred quasar distance, the propagation of individual jet clumps away from the quasar can appear to be faster than the speed of light!

From this it is clear that apparent super-luminal propagation is possible whenever the propagation speed v > 0.707c.

The result shows a remarkable 'cosmic web' in the overall large-scale structure (LSS) of the universe.

The inability to produce light is indeed what makes WIMPs a candidate for dark matter.

Indeed, a key question is whether gravity might be strong enough to stop and even reverse the expansion, much as occurs when an object is launched with less than Earth's escape speed.

As the universe thus eventually closes back on itself, this is known as a "closed" universe.

But an alternative, indeed more general and physically more appropriate perspective, is that this redshift is actually just a consequence of the expansion of space itself!

Note that, as implied by the expansion (30.22), all the models converge to the simple linear Hubble law (purple line) at modest distances, d << dH = c/H0.

But in one the greatest surprises of modern astronomy, and indeed of modern science, such data points were found to generally lie below the black curve that represents a nearly-empty universe.

Then, after Hubble's discovery that the universe is not static but expanding, Einstein completely disavowed this cosmological constant term, famously calling it 'his greatest blunder'.

Thus, in contrast to the previous case of constant expansion for an empty universe with Ωm = ΩΛ = 0, for a dark-energy-dominated, flat universe with ΩΛ = 1, the expansion actually accelerates exponentially, with an e-fold increase each Hubble time!

If instead, the initial Ω had been even slightly above unity, the fledgling universe would have recollapsed as a tiny, closed universe.

After working hard to reduce electronic and other possible sources of static, they eventually concluded the noise was actually coming from the sky.

This can be considered as the present-day "temperature of our universe".

Much as measurement of seismological waves generated in an earthquake provide information on the interior structure of the Earth, these measures of CMB fluctuation power peaks provide information on the pre-recombination evolution of the universe.

The photons from this recombination era thus were suddenly free to propagate through the universe, becoming redshifted by its expansion to form the CMB we observe today.

While Helium is synthesized in stars, it turns out that most of the Helium in the universe today was actually formed in the first few minutes or so after the Big Bang, when the temperature was several billion degrees (109K).

Cores of stars are thus relatively low-temperature "slow cookers" of He compared to the rapid nucleosynthesis in the first few minutes of the Hot Big Bang.

In this particle era, the universe was thus very nearly symmetric between matter and anti-matter.

The upshot is that the Planck era is at the very frontier, where of our current physical understanding is untested and breaks down into a "quantum foam".

The initially tiny physical scale of these fluctuations was amplified by inflation to much larger structures that we see today in the angular spectrum of fluctuations of the CMB.

In the ghostly world of quantum mechanics, electrons are themselves not entirely discrete particles, but rather, much like light, can also have a 'wavelike' character.

This leads to what is known as 'natural broadening' of spectral lines.

The Lyman series transitions all fall in the ultraviolet (UV) part of the spectrum, which due to UV absorption by the earth's atmosphere is generally not possible to observe from ground-based observatories.

At high temperatures, the energy of collisions can become sufficient to overcome the full binding energy of the atom, allowing the electron to become free, and thus making the atom an ion, with a net positive charge.

This large number of states acts like a kind of "attractor" for the ionized state.

Because an isolated electron has no way to store both the energy and momentum of the incoming light, it cannot by itself absorb the photon, and so instead simply scatters, or redirects it.

An everyday analogy is blowing into a whistle vs. just into open air.

This so-called "Eddington-Barbier relation" states that when you peer into an opaque radiating gas, the emergent intensity you perceive is set by the value of the blackbody function at the location of unit optical depth along that ray.

Comparison of the final form of (D.4) with the simple discussion of surface flux in part I shows that we can identify what we've been calling the stellar "surface" as the layer where the optical depth (R) 2/3.

And since the Greeks decided to give dimmer stars a higher magnitude, we find that magnitude scales with the log of the inverse flux.

As a mneumonic, this is cast as a 10-digit "telephone number", with the 3-digit "area code" representing the 3 steps of 10-5 from us down to the nucleus, and 7-digit main-number representing 7 key steps to the scale of the universe.

The full sequence of steps over this span thus looks something like a 10-digit phone number with area code: 555-711-2555.

The apparent angular size that object subtends in our overall field of view is then used intuitively by our brains to infer the object's distance, based on our extensive experience that a greater distance makes the object subtend a smaller angle.

The key point here is that the parallax angle shift of your finger, which results from switching perspective from one eye to the other, exactly fits the apparent angular separation between your own mirror-image eyes.

The parallax for even the nearest star is less than an arcsec, implying stars are all at distances more (generally much more) than a parsec.

Integration over a full sphere shows that there are 4π steradians in the full sky.

This is a profoundly important equation in astronomy, and so you should not just memorize it, but embed it completely and deeply into your psyche.

Thus we see that the Sun emits the power of about 4 x 10^24 100-watt light bulbs! In common language this corresponds to four million billion billion, a number so huge that it loses any meaning.

Each photon carries a discrete, indivisible 'quantum' of energy that depends on the wave frequency.

The reason there are so many stars in this main-sequence band is that it represents the long-lived phase when stars are stably burning Hydrogen into Helium in their cores.

Within quantum mechanics, it thus has an associated 'fuzziness' in position, characterized by its De Broglie wavelength λ = h/p, where h is Planck's constant.

This eventually leads to a catastrophic core collapse, halted only when the electrons merge with the protons in the Iron nuclei to make the entire core into a collection of neutrons.

The final, merged black hole was inferred to be about 62 M, with the extra 3M converted to energy in the emitted gravitational wave, which for brief, few msec of the merger represented some 50 times the luminosity of all the stars in the observable universe!

You and I and everyone on Earth are here today because our source material happened to stem from the equatorial regions of the proto-solar core, with too much angular momentum to fall into the Sun itself.

In the context of the prevailing idea that such gas giants should only form beyond the ice line, this detection was a real surprise.

This comparison illustrates why these flat rotation curves came as a surprise.

The star S0-2 has the shortest period, 16.7 years, and so has been tracked over more than a full orbit.

Einstein's limit really applies to how fast objects can travel relative to space, but that space itself can expand at a speed faster than light!

But in one the greatest surprises of modern astronomy, and indeed of modern science, such data points were found to generally lie below the black curve that represents a nearly-empty universe.

This can be considered as the present-day "temperature of our universe".

The initially tiny physical scale of these fluctuations was amplified by inflation to much larger structures that we see today in the angular spectrum of fluctuations of the CMB.