Widdekind

re-EDIT: To emphasize the 'gravity-well' effect, assume zero velocity; and, assume that some emission line implies [math]\lambda_e[/math], whilst [math]\lambda_o[/math] is known. Then: [math]\frac{1 - \frac{R_{S,o}}{r} }{ 1 - \frac{R_{S,e}}{r}} (1+z) \approx\frac{\lambda_o}{\lambda_e} [/math] showing that light, blue-shifting in-bound at the observer, much increase the inferred red-shift; and, to account for non-cosmological red-shifting out-bound at the emitter, "before the light is loosed-and-let-to-fly out into the Hubble Flow", one must reduce their inferred cosmological red-shift value. Could this be relevant for photons emitted near compact objects, e.g., GRBs ?? However, for GRBs, I understand that [math]\lambda_e[/math] is unknown; and, that known cosmological red-shifts are obtained, separately, from their host galaxy. Thus, [math] \lambda_e \approx \left( \frac{\lambda_o}{1+z} \right) \frac{1 - \frac{R_{S,e}}{r} }{ 1 - \frac{R_{S,o}}{r}}[/math] showing that as [math]r_e \rightarrow R_{S,e}[/math], then [math]\lambda_e \rightarrow 0[/math], i.e., the radiation is "harder than it looks". And, indeed, long-period GRBs typically are observed to have softer radiation (~1/3 me c2), but be BH-generated; whereas, short-period "short hard" GRBs typically are observed to have harder radiation (~1/2 me c2), but may be NS-merger generated (Mazure & Basa. Exploding Superstars; Bloom. What are GRBs ?). That picture appears to be completely self-consistent, since BH-generated l-GRBs emit their light closer to the Schwarzschild radius. Note, tho, that equally plausibly, humans on earth only observe GRBs, when looking "down the bore-sight", straight towards a pole, of the central compact object. Thus, matter in-falling towards said central accretor, would be moving radially away from earth -- and, hence, any light emitted "back up-gravity-well", towards earth, would be redshifted. Thus, matter infalling faster, towards a BH, during an l-GRB, could also account for the inferred redshifting of the radiation (assumed to be from pair production, at 511 KeV), relative to a somewhat slower infall, towards the NSs, in an s-GRB. Therefore, matter infalling towards the central compact object, at a considerable fraction of the speed-of-light; or, radiating matter residing, close to said compact object (~1.5-2 RS), could both account, jointly or severally, for the relative redshift, of l-GRBs, compared to s-GRBs.

According to Eric V. Linder's First Principles of Cosmology, pg.88, [math]1+z = \frac{\left( \vec{k} \circ \vec{u} \right)_e}{\left( \vec{k} \circ \vec{u} \right)_o}[/math] where [math]\vec{k} \circ \vec{u} = g_{\mu \nu} k^{\mu} u^{\nu}[/math]. Yet, when that four-dot-product is taken, the Minkowski (locally flat-space) metric tensor seems to be utilized, such that (for a radially-inbound [math]e \rightarrow o[/math] photon): [math]1+z = \frac{\lambda_o}{\lambda_e} \frac{\gamma_e \left( 1 + \beta_{r,e} \right)}{\gamma_o \left( 1 - \beta_{r,o} \right)}[/math] What would happen, if you considered cosmologically-distant light-sources, deep down in a gravity well? If, as a first approximation, one were to assume a Schwarzschild space-time, is not the Schwarzschild Metric: [math] \left( \begin{array}{cccc} 1-R_S/r & 0 & 0 & 0 \\ 0 & -1 / (1- R_S/r) & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right)[/math] ? Then, the four-dot-product time-time components [math]1 \rightarrow 1-R_S/r[/math]; and, the space-space components [math]\beta_r \rightarrow \beta_r / (1-R_S/r)[/math], so that: [math]1+z = \frac{\lambda_o}{\lambda_e} \frac{\gamma_e \left( (1- \frac{R_{S,e}}{r}) + \frac{\beta_{r,e}}{1-R_{S,e}/r} \right)}{\gamma_o \left( (1- \frac{R_{S,o}}{r}) - \frac{\beta_{r,o}}{1-R_{S,o}/r} \right)} \approx \frac{\lambda_o}{\lambda_e} \frac{ \left( (1- \frac{R_{S,e}}{r}) + \beta_{r,e} (1+R_{S,e}/r) \right)}{\left( (1- \frac{R_{S,o}}{r}) - \beta_{r,o} (1+R_{S,o}/r) \right)} = \frac{\lambda_o}{\lambda_e} \frac{ \left( 1 + \beta_{r,e} - \frac{R_{S,e}}{r} (1 - \beta_{r,e}) \right)}{\left( 1 - \beta_{r,o} - \frac{R_{S,o}}{r} (1 + \beta_{r,o}) \right)}[/math] to lowest orders, in velocities and radii terms. For the Schwarzschild terms, the time-originating piece dominates the space-originating piece, and actually reduces the redshift, if the emitter is deep down in a gravity well. Could that be correct ?? Is not the RS of our galaxy, of [math]\approx 10^{12} M_{\odot}[/math], thusly about one-third-light-year ? If so, then for us, out ~30Kly from our galactic core, the effect would be O(10-5). EDIT: To emphasize the 'gravity-well' effect, assume zero velocity; and, assume that some emission line implies [math]\lambda_e[/math], whilst [math]\lambda_o[/math] is known. Then: [math]\frac{1 - \frac{R_{S,o}}{r} }{ 1 - \frac{R_{S,e}}{r}} (1+z) \approx\frac{\lambda_o}{\lambda_e} [/math] showing that light, blue-shifting in-bound at the observer, much increase the inferred red-shift; and, to account for non-cosmological red-shifting out-bound at the emitter, "before the light is loosed-and-let-to-fly out into the Hubble Flow", one must reduce their inferred cosmological red-shift value. Could this be relevant for photons emitted near compact objects, e.g., GRBs ?? However, for GRBs, I understand that [math]\lambda_e[/math] is unknown; and, that known cosmological red-shifts are obtained, separately, from their host galaxy. Thus, [math] \lambda_e \approx \frac{\lambda_o}{1+z} \frac{1 - \frac{R_{S,e}}{r} }{ 1 - \frac{R_{S,o}}{r}}[/math] showing that as [math]r_e \rightarrow R_{S,e}[/math], then [math]\lambda_e \rightarrow 0[/math], i.e., the radiation is "harder than it looks". And, indeed, long-period GRBs typically are observed to have softer radiation, but be BH-generated; whereas short-period "short hard" GRBs typically are observed to have harder radiation, but may be NS-merger generated. That picture appears to be completely self-consistent, since BH-generated l-GRBs emit their light closer to the Schwarzschild radius.

'Rampart Craters' on Mars show signs of mud-flows, over-land, w/o ejecta spray, through the air. They may occur when impactors fall on ice-saturated ground, which ice absorbs allot of energy in melting & liquifying. And, Mars has more than twice as strong a surface gravity as our moon. Thus, could one arrange for "sloppy landings", of de-orbited asteroids, onto prepared "ice-mud-flats", e.g., on the floors of large pre-existing craters ??

Astrophysical jets are cosmologically common, occurring in systems as diverse, as forming proto-stars (in Bok Globules): to post-stellar remnants, in X-ray Binaries: Now, jets are 'anti-symmetric', since they travel in opposite directions, both being out-bound. Thus, reversing the rotation of the disk-and-central-object, would not affect the jets. Such an 'anti-symmetry' is characteristic of dipoloidal magnetic fields, whose 'looping' field lines converge down into the disk, from one pole; and diverge up away from the disk, towards the other pole. That 'odd symmetry' of the magnetic field, implies that ions above the disk would be accelerated upwards, whereas ions below the disk, would be accelerated downwards -- to wit, both outbound. Thus, provided that the accretion disk rotates in the same sense, as the central magnetic-field-generating-object, then dipoloidal magnetic fields 'have the required character' for the central engine powering jets.

If, in the FRW metric, the 'interval' is: [math]ds^2 = (c \, dt)^2 - R(t)^2 \left( \frac{dr^2}{1 - K r^2} + r^2 \left( d \theta^2 + ( sin(\theta) d\phi )^2 \right) \right)[/math] And if, in GR, that 'interval' is derived from the 'metric' tensor, via: [math]ds^2 = g_{\mu \nu} ds^{\mu} ds^{\nu}[/math] And if [math]ds \equiv \left[ c dt, dr, d\theta, d\phi \right][/math]; then, why wouldn't the 'metric' tensor be: [math] \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & R^2 / 1- K r^2 & 0 & 0 \\ 0 & 0 & R^2 r^2 & 0 \\ 0 & 0 & 0 & R^2 r^2 sin(\theta)^2 \end{array} \right)[/math] or, if [math]ds \equiv \left[ c dt, R dr, R r d\theta, R r sin(\theta) d\phi \right][/math] (to put all the physical distances into the interval); then, why wouldn't the 'metric' tensor be: [math] \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 / 1- K r^2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)[/math] ?

Metals can cool the clouds, in the ISM, from which stars form. And, the properties of those clouds could be characteristic, of the outer envelopes, of stars: Indeed, for a given 'core operating power', and hence, ultimately, luminosity, more metals makes stars colder (redder), but bigger -- shifting the MS to the left, on the HRD: (source: mnsu) Indeed again, more metals in the stellar mixture modifies post-MS evolution, shifting He-core-burning stars, from the 'Horizontal Branch' (low Z), leftwards, to the 'Red Clump' (high Z) (access-science). The bigger, bloated, higher metallicity stars also (therefore?) rotate less rapidly; and, have higher mass-loss rates (~Z1/2) (supersci). In turn, these properties of stars, during their lives on the MS, affect their post-stellar evolution off of the MS -- in particular, the SNe / HNe / GRBs of high-mass stars: That is, low-metallicity stars are smaller, more compact, have hotter envelopes, and rotate more rapidly. They also have higher core-masses (He thru Fe), which cores re-attract more 'fall-back' material, during SNe. And, indeed, faster-rotating & more-massive remnants, re-accreting more material, could account for the 'collapsar' model of (long-type) GRB generation, during the initial phases, of the HNe, of hyper-massive stars, in the early universe.

Escaping radiation might be rather red-shifted ?

Low-mass stars are 'slow all the way around' -- they not only burn thru their initial H fuel more slowly, but also form more slowly, from their proto-stellar cloud precursors: QUESTION -- If cosmic 'Reionization' occurred, comparatively quickly, billions of years ago, would all that ionizing radiation, from high-mass 'Pop. III' stars, have quenched the birth of low-mass-and-slow-assembling stars ? For example, 2 billion years later, low-mass-and-slow-assembling galaxies were similarly quenched by Quasars, during secondary (He) reionization.

...small arcs, each of which in-falls into the BH, due to practicalities, of initial emission angle/direction, and/or interaction, with in-falling matter, streaming thru the radiation 'belt' ?

Most of the mass, of normal atomic matter, is associated with the nucleons, in the atomic nuclei. And, within those nucleons, most of the mass (~930 of 940 MeV), is associated with the gluon 'glue' keeping the 'naked' quarks confined. Now, in an ultra-compact object, the extreme compressions, caused by gravity, could keep quarks confined, as a substitute for gluon Color Force interactions. And, with the Color Force, bond-energies decrease, with decreasing spatial separations, between the interacting quarks. Thus, if gravity 'substituted' for Color Force confinement, of quarks, in nucleons, in ultra-compact objects; then, Color Force interactions might decrease, dramatically decreasing the amount of mass-energy in quark-confining glue. To wit, might extreme gravitation compression, cause an effective reduction in mass ??

Would a BH, accreting matter from an orbiting accretion disk, be orbited, close to or at its EH, by a "ring of light" ?? Please ponder the "head-light effect", whereby accelerated charged particles, moving relativistically, emit hard radiation, focused in their forward direction. Thus, as ions are dragged down into the BH, they will emit "headlights" of radiation, focused forward, around the BH. Would such radiation combine, to produce an annulus of orbiting photons, or "light ring", around the BH ??

So, seemingly, small WDs will be "spent He-cores", billions of years far into our future, once those slow-burning, M-class 'red dwarves' ([math]M_i < 0.4 M_{\odot}[/math]) finally fuse their last H atom. (That is an important caveat) EDIT: this link looks relevant

(source: Lecture 9) According to Duric's Advanced Astrophysics, under human-familiar kinds of conditions (low Temperature, Pressure, Density), neutrons spontaneously decay, into protons & electrons, b/c the rest-masses of those daughter particles, is less than that of the neutron. But, in NS, the ultra-compaction elevates the QM Fermi Energy [math]\frac{\har^2}{2 m \Delta X^2}[/math] to such a dramatic degree, that [math]E_F > \Delta M c^2[/math], and the decay reaction is driven in reverse. Now, neutrons, whose quark content is ddu, have excited spin-and-magnetic-moment states, e.g. [math]\Delta^0[/url]. Could the ultra-compaction create "Delta stars" ?

The magnetization, of a uniformly magnetized sphere, is derivable from [math]\vec{B} = \frac{2}{3} \mu_0 \vec{M}[/math] and [math]\vec{m} = \frac{4 \pi}{3} R^3 \vec{M}[/math]. Now, a NS exists, at the nuclear density of nn = 1.4 x 1044 m-3; and, is composed of neutrons, whose magnetic moments are mn = (-)0.97 x 10-26 J T-1. Thus, the quantum maximum volume magnetization density, of a NS, is M = mn nn. And so, from observed surface magnetic fields, we can define a 'magnetization fraction': [math]B_{obs} \equiv f \left( \frac{2}{3} \mu_0 \right) m_n n_n \approx f \times 90 \; GT[/math] Thus, the 'quantum critical' magnetic field, for NS, is ~1015 G. All observed young Magnetars have inferred field strengths of >1014 T, implying 'magnetization fractions' f > 10% (see following figure). Thus, a significant fraction, of ultra-magnetized NS, are 'unpaired neutrons', whose spins, and magnetic moments, are aligned, generating star-sized, nuclear-strong, magnetic fields. Now, Magnetars generate immense quantum magnetic fields: And, combined with their relatively rapid rotation, they emanate intense 'magnetic dipole radiation'. These emanations come in bursts & flares, possibly when: Such a scenario sounds a little like a 'magnetic-field-line-reconnection' event, during a solar flare. QUESTIONS: What could cause neutrons, which normally 'prefer' to be paired with spin-opposite partners, to spin-align, in Magnetars ? Could Quantum Spin-Flips, a manifestation of the quantum 'jump' phenomenon, account for the sudden 'glitch' transitions, and associated flares ? Perhaps NS are a little like ferro-magnets, with 'domains' of spin-aligned matter, which 'quantum jump' en masse, to more stable spin states ? (source: Kondratiev 2009) Note that, the SN, of super-massive stars, which produce NS, can also produce 'Pulsar planets', evidently from 'fall back' of material, which fails to escape the deep gravity wells, of the progenitor-and-resulting-remnant. Might, then, there be 'White Dwarf planets', orbiting other post-stellar remnants; or, is the necessary 'fall-back' of material characteristic only of the more intense gravity, which generates NS ?

So, seemingly, small WDs are "spent He-cores", perhaps up to [math]\approx 0.5 M_{\odot}[/math].

Core-Contraction powers GB luminosities ? "Gravitational Luminosity", the power provided by core contraction, could account, for observed post-MS luminosities, with modest core-contraction infall velocities: [math]U_g \approx -\frac{3}{5} \frac{G M_c^2}{R_c}[/math] [math]L_g \equiv \frac{dU_g}{dt} \approx \frac{3}{5} \frac{G M_c^2}{R_c^2} \dot{R_c} \approx \left( 10^8 L_{\odot} \right) \left( \frac{M_c}{M_{\odot}} \right)^2 \left( \frac{R_c}{0.1 R_{\odot}} \right)^{-2} \dot{R_c}[/math] Thus, as a sun-like star evolves off the MS, core-contraction infall velocities of ~10 nm / s could account, for observed proto-RGB luminosities. And, even the peak RGB luminosity [math]\approx 10^4 L_{\odot}[/math], when the He-core has contracted to WD size [math]\approx 10^{-2} R_{\odot}[/math], at the RGB-to-HB "bounce" on the HRD, core-contraction infall velocities of ~1000 nm / s could still account, for observed peak RGB emission power: After the "bounce", He-core ignition occurs, and the now-WD-sized, He-fusing-to-C core stabilizes the star, onto the HB. Then, once core He is depleted, the core-contraction process repeats, and the now-WD-sized core is (partially) "trash-compacted" yet again. And again, similar rates of core-contraction could account, for observed AGB luminosities. every GB trajectory represents core-contraction & "trash-compaction" ? every "bounce" back towards MS (e.g., HB) represents core-re-ignition ? Massive stars traverse "to-and-fro" on HRD, (1) evolving away from MS, perhaps powered by core-contraction; and then (2) "bouncing" back towards MS, perhaps powered by a stabilized-and-re-ignited core, until the next core "flare-out-and-failure" -- at which point the cycle may repeat. Thus, could the cyclical migrations, of massive stars, "back-and-forth" across the HRD, be "counted like tree-rings", to infer the internal "core compaction state", and commensurate core-fusion processes ? At the present epoch, most massive stars SN in their red-giant phase, well off-and-to-the-right of the MS, on the HRD (Mazure. Exploding Superstars, p.49; Wheeler. Cosmic Catastrophes, p.261): Yet, long long ago, most massive stars, of much lower metallicity, may have SN'd as blue-giants, near the MS, like SN1987a: (source: SN1987a) Could it be, that red-giant phase SN occur at a core-re-ignition "bounce" event, and so represent Thermo-Nuclear SN (e.g., SNIa) ? And/or, that blue-giant phase SN occur at a core-failure-and-flare-out event, and so represent a true neutronizing-and-neutrino-generating Core-Collapse SN (e.g., SNII) ? AGB phase, of middle-mass stars, represents repeated "bounces", of periodic C-core fusion flashes, whose prodigious power exceeds stars' gravitational confinement capacity, so expelling envelope, into burgeoning PNe ?? AGB 'thermal pulses' typically last 10-100 kyr, begun by few kyr of C-core fusion 'flash', whose power cannot be contained by over-lying envelope (cf. Tokamaks), which is blown back out towards space. Pressure drop quenches C-core fusion, and envelope ballistically expands, cools, and re-collapses, re-initiating subsequent cycle. However, with each fusion 'flash', some overlying envelope is completely ejected, and remainder, having been blown out to ever higher altitudes, comes "crashing back down" ever harder, compressing the core ever more, thereby exaggerating & escalating subsequent C-core fusion flashes. These un-harnessed fusion flashes also blow C-core material out into ejected envelope: Is this an accurate understanding, of the AGB thermal pulse process ?? What would be a good-yet-accessible book, on star fusion processes ?? the carbon core contracts, and Temperature increases, but not enough to ignite carbon or oxygen...[fusion] flashes, in a shell surrounding the core, ejecting an envelope (uvic) Thermal Pulses of AGB stars (mnsu). Note that the initial luminosity 'spikes' resemble that of SN1987a.

According to reference, Does that mean, that some WDs, are composed of He (as the spent cores of low mass stars), not CNO ??

http://astro.berkeley.edu/~basri/bdwarfs/

"Dark Luminosity" -- non-radiative, "Kinetic Power Luminosity", of stellar winds, becomes 1-10% of radiative stellar Luminosity, for high mass stars ? [REVISED] CNO-cycle MS stars, of [math]\approx 2-20 M_{\odot}[/math], typically eject ~80% of their initial ZAMS masses, into surrounding circum-stellar (pre-PNe ?) nebular formations, over their lifetimes: fig.1-- Total integrated mass-loss thru stellar evolution (source: Kalirai 2010) Thus, the mass-loss rates, of these 'high mass' stars, is approximately proportional, to their Luminosities: [math]\Delta M \propto M[/math] [math]\dot{m} \approx \frac{\Delta M}{\tau} \approx \frac{M}{\left( \frac{M}{L} \right) } \approx L[/math] And, these mass-loss rates, represent a 'kinetic power', of [math]\frac{1}{2} \dot{m} v_w^2[/math]. Thus, the stellar-winds, of 'high mass stars', represent a "dark luminosity", whose ratio to visible, radiative luminosity, is: [math]\frac{L_w}{L_r} = \frac{1}{2} \frac{\dot{m} v_w^2}{L_r} = \frac{1}{2} \frac{\dot{m}_w \beta_w^2}{\dot{m}_r} \rightarrow \frac{1}{2} \frac{\Delta M_w}{\Delta M_r} \beta_w^2 \approx \frac{1}{2} \frac{0.8 M}{10^{-3} M} \beta_w^2[/math] For wind-velocities of a few thousand km/s, nearing 1% the speed-of-light, the above ratio approaches 1%. Thus, for high-mass, CNO-cycle MS stars, star-winds represent a "dark luminosity", ~1% of visible radiative luminosity, whose driving power source, must derive, from the fusion, in the operating star-core. This is the 'opposite', of NS-generating, core-collapse SNe, wherein the explosive 'super-winds' carry off 100x more energy, than the visible radiative display (Kirshner. Extravagant Universe, p.36). And, for super-massive stars (O,B, [math]> 20 M_{\odot}[/math]), whose radiative luminosities approach [math]10^6 L_{\odot}[/math], and whose mass-loss rates approach [math]10^{-4} M_{\odot} yr^{-1}[/math], their "kinetic luminosities" can exceed 10% of radiative luminosity. Such suggests, that the kinetic luminosities, embodied in stellar wind mass loss, become increasingly important, for more & more massive stars. And, that qualitatively resembles the increasing importance, of radiation pressure, in the same. Indeed, the ratio of radiation-to-thermal pressure, inside stars, scales roughly as: [math]\frac{P_r}{P_{th}} \approx 10^{-4} \left( \frac{M}{M_{\odot}} \right)^2[/math] (Mazure. Exploding Stars, p.131) Thus, as stars approach the Eddington Limit ([math]\approx 120 M_{\odot}[/math]), radiation pressure begins to photo-evaporate the star (ibid., p.38). If so, then star-winds may be driven by radiation pressure (?). 'Low-Mass' stars ([math] < \frac{1}{2} M_{\odot}[/math]) Indeed, extrapolating from fig.1, mass-loss is negligible for 'low-mass' stars [math] < 0.5 M_{\odot}[/math]. Moreover, according to observations "of galactic open cluster(s)", remnant mass, as a function of initial stellar mass, scales roughly as [math]M_{WD} \approx 0.5 + 0.07 M_i[/math] (with masses measured in solar units): fig.2 -- "big dots and solid curve report the Weidemann (2000) empirical relation based on the mass estimate of white dwarfs in Galactic open cluster" (source: Buzzoni) Obviously, however, [math]M_{rem} \leq M_i[/math], so that below [math]\approx 0.5 M_{\odot}[/math], mass-loss is mathematically negligible, and [math]M_{rem} \approx M_i[/math]. Moreover, these low-mass stars do not form White Dwarves: And, indeed, [math]< 0.5 M_{\odot}[/math], the stellar mass-luminosity relation changes markedly: Now, all stars [math]< 2 M_{\odot}[/math] generate core power, by the same PP cycle. Such suggests, that even low-mass stars "should" obey the same mass-luminosity trend, as middle-mass stars: (source: unl) If so, then low-mass stars "ought" to be both bigger ([math]\approx R_{\odot}[/math]), and cooler ([math]\approx 1000 K[/math]), i.e., shifted rightwards on the HRD. That the transition, from low-mass, to middle-mass, behavior occurs, at an effective photo-spheric temperature, of [math]\approx 5000 K[/math], suggests the importance of H ionization (cf. CMBR) and plasma opacity, to burgeoning radiation pressure. Now, output luminosity is a direct measure of core operating power, which is determined solely by star mass -- "the evolution of a star is regulated, by one parameter alone, and that is its mass" (Mazure. Exploding Superstars, p.39) And, logically, core operating power is proportional, to the product, of core mass x specific power, [math]L \propto M_{core} \times \epsilon \equiv \alpha M \times \epsilon[/math], where we have introduced a 'core-mass-fraction' [math]\alpha \equiv M_{core} / M[/math]. Thus, for PP-cycle stars [math]< 2 M_{\odot}[/math], w.h.t.: [math]M^4 \approx L \approx \alpha M \epsilon[/math] [math]\therefore \alpha \epsilon \approx M^3[/math] Furthermore, for most MS stars, the core-mass-fraction is quasi-constant, ~10% (Mazure, p.42,134). Thus, for the PP-cycle, [math]\epsilon \approx M^3[/math]. Yet, for low-mass stars, [math]M^{2.3} \approx L \approx \alpha M \epsilon \approx \alpha M^4[/math] [math]\therefore \alpha_{< 0.5 M_{\odot}} \approx M^{-1.7}[/math] To wit, low-mass stars may be increasingly 'core-dominated' (cf. over-large iron core of Mercury). Indeed, according to those calculations, minimum-mass stars [math]\approx 0.1 M_{\odot}[/math] could have core-mass-fractions of [math]\approx 1[/math]. Thus, the smallest stars might be "all core", which could be construed as consistent, with their being both "too small-and-dense", as well as "too hot" (since their core would be "too close" to stellar surface), and "too bright" (compared to the expected PP-cycle M-4 decline). This site does seem to say, that core-mass-fraction does decrease, with increasing star mass, assigning 0.6 solar-mass core to 0.8 solar-mass stars (3/4), 1.5 solar-mass cores to 9 solar-mass stars (1/6), and then asymptoting towards ~10%, above 10 solar masses (1/10). Is this reasonable? What would this say, about (super-)giant planets ?? Can this be construed, as consistent, with the claim, that as star mass increases, increasing temperatures, and hence radiation pressures, increasingly "blow more of the star up-out-and-away" (mechanically, viewing stars as 'engineering structures', more of the star's mass is 'diverted', from the core, to the outer envelope, where its weight will gravitationally confines the core) ??

0.1% rule (source: planet facts)

"Dark Luminosity" -- non-radiative, "Kinetic Power Luminosity", of stellar winds, becomes 1-10% of radiative stellar Luminosity, for high mass stars ? CNO-cycle MS stars, of [math]\approx 2-20 M_{\odot}[/math], typically eject ~80% of their initial ZAMS masses, into surrounding circum-stellar (pre-PNe ?) nebular formations, over their lifetimes: Total integrated mass-loss thru stellar evolution (source: Kalirai 2010) Thus, the mass-loss rates, of these 'high mass' stars, is approximately proportional, to their Luminosities: [math]\Delta M \propto M[/math] [math]\dot{m} \approx \frac{\Delta M}{\tau} \approx \frac{M}{\left( \frac{M}{L} \right) } \approx L[/math] And, these mass-loss rates, represent a 'kinetic power', of [math]\frac{1}{2} \dot{m} v_w^2[/math]. Thus, the stellar-winds, of 'high mass stars', represent a "dark luminosity", whose ratio to visible, radiative luminosity, is: [math]\frac{L_w}{L_r} = \frac{1}{2} \frac{\dot{m} v_w^2}{L_r} = \frac{1}{2} \frac{\dot{m}_w \beta_w^2}{\dot{m}_r} \rightarrow \frac{1}{2} \frac{\Delta M_w}{\Delta M_r} \beta_w^2 \approx \frac{1}{2} \frac{0.8 M}{10^{-3} M} \beta_w^2[/math] For wind-velocities of a few thousand km/s, nearing 1% the speed-of-light, the above ratio approaches 1%. Thus, for high-mass, CNO-cycle MS stars, star-winds represent a "dark luminosity", ~1% of visible radiative luminosity, whose driving power source, must derive, from the fusion, in the operating star-core. This is the 'opposite', of NS-generating, core-collapse SNe, wherein the explosive 'super-winds' carry off 100x more energy, than the visible radiative display (Kirshner. Extravagant Universe, p.36).

If a state prepared as spin "up", is subsequently quantum-Measured, along an orthogonal direction (e.g., "left-right"), the probability is 1/2 that the state will be observed, in either of the two orthogonal, and mutually anti-parallel, states... yes? Thus, the spin "up" state, can be construed, as a SP, of "left" and "right". That's the impression I get, from Kenneth Ford's 101 Quantum Questions.

Dividing the Planck Density (5e96), by the current Cosmic Critical Density (1e-26), and taking the cube-root, yields ~e41. Thus, at a red-shift of z ~ 1041, our Observable Universe would have been compacted to Planck Density. Is that when/where current physical theory breaks down?

At a given pressure, inside stars, for a given inward gravity, electrons, being less massive, experience less inward pull, despite their similar thermal energies, and higher velocities. I read in a book, that this means stars shed a few electrons, until their slight positive charge affords electrons a balancing EM attraction. The following equations attempt to mathematically model said effect. First, from the Virial Theorem [math]<U> = -2 <K>[/math], applied separately to the electronic, and ionic, components, of the stellar plasma, w.h.t. [math]-\frac{G M m_e}{R} - \frac{Q e}{4 \pi \epsilon_0 \, R} = -2 \left( \frac{3}{2} k_B T_e \right)[/math] [math]-\frac{G M <A> m_p}{R} + \frac{Q <Z> e}{4 \pi \epsilon_0 \, R} = -2 \left( \frac{3}{2} k_B T_i \right)[/math] Now, assuming thermalization of electrons & ions, s.t. [math]T_e \approx T_i[/math], we can equate these expressions, s.t.: [math]-\frac{G M m_e}{R} - \frac{Q e}{4 \pi \epsilon_0 \, R^2} = -\frac{G M <A> m_p}{R} + \frac{Q <Z> e}{4 \pi \epsilon_0 \, R}[/math] which, after algebraic manipulations, yields (in Coulombs): [math]Q = \left( \frac{4 \pi \epsilon_0}{e} \right) \left( G M m_p \right) \frac{<A>}{<Z>+1} \approx 300 \left( \frac{M}{2 M_{\odot}} \right) \frac{<A>}{<Z>+1}[/math] since NS are [math]\approx 2 M_{\odot}[/math], characteristically (Kirshner. Extravagent Universe, p.36). For a pure H plasma, [math]A = Z = 1[/math], so that the 'plasma fraction' is ~1/2; whereas for a pure (and fully ionized) Fe plasma, characteristic of massive pre-collapse star cores (at [math]T \approx 300 keV[/math]) w.h.t. [math]<A> \approx 2 <Z>[/math], so that the 'plasma fraction' is ~2. Even in such a case, that's only 0.6 m-mol of electrons, out of an entire NS worth of mass. Now, a spinning charged sphere generates a magnetic moment, and dipole-like magnetic field, of characteristic surface-strength [math]B \approx \frac{\mu_0}{4 \pi} \frac{m}{R^3}[/math], where the magnetic moment [math]m \approx \frac{Q}{2 M} L = \frac{1}{5} Q R^2 \omega[/math]. Thus, w.h.t. (in Tesla): [math]B \approx \frac{1}{5} \frac{\mu_0}{4 \pi} \frac{Q \omega}{R} \approx 8 \times 10^{-6} \left( \frac{M}{2 M_{\odot}} \right) \left( \frac{R}{7 km} \right)^{-1} \left( \frac{\omega}{716 Hz} \right)[/math] even using the most rapidly rotating pulsar observed to date. Thus, even a 1ms pulsar, would not be expected, according to these equations, to generate a surface magnetic field, in excess of 0.1G, through this method.

Please ponder (neutral-)pion decay, [math]\pi^0 \rightarrow \gamma + \gamma[/math]. The resulting pair of photons are quantum-Entangled, from the spontaneous pion decay and coupled creation of the photon pair, up until the first quantum-Measurement performed upon the pair (Ford. 101 Quantum Questions). By symmetry, each photon of the pair should (1) do "everything it can", i.e., "go left, go right; spin left, spin right"; and (2) do "everything opposite what the other guy does", i.e., "I go left, when he goes right; I spin left, when he spins right, etc." The latter (2) ensures conservation of (zero) total linear & angular momenta (zero, both before, and after, the spontaneous pion decay). Now, the full, quantum-Entangled, Wave-Function super-poses such "equal-but-opposite" states, with a minus-sign, not a plus-sign. And, that anti-phase addition is quantum-physically important (Ford. 101 Quantum Questions). Cannot one visualize that minus sign, as essentially saying, that the super-posed photon states, are 180-degrees out of phase, w.r.t. the vector-potentials [math]\vec{A}_{1,2}[/math] that generate those photons ? (Recall, the direction of [math]\vec{A}[/math] defines a photon's direction of polarization; and, that vector-potential generates both [math]\vec{E} = -\frac{\partial \vec{A}}{\partial t}[/math] and [math]\vec{B} = \nabla \times \vec{A}[/math].) If so, then (neutral-)pion decay generates two equal-but-opposite photons, broken down into four co-coupled quarters-of-Wave-Functions, two of which are pro-phase ([math]\vec{A}[/math] "zigs then zags"), and the other two of which are anti-phase ([math]\vec{A}[/math] "zags then zigs"). If so, then, making a monetary analogy, (1) each photon "dollar" is generated in four "quarters", two of which go left (and two go right), and two of which spin left (and two spin right), with two "zigging & zagging" and the other two "zagging & zigging", so that each "dollar" has zero net linear, and zero net angular, momentum, and zero net vector potential (zero net "zig plus zag", as it were); (2) each photon "quarter" is cross-coupled, to anti-corresponding "quarters", of the other photon: fig. 1 -- the first photon is symbolized by a F-16; the second photon is symbolized by a Sukoi-35; direction of travel is directly indicated, and denoted with capital letters (L,R); direction of spin is indicated by upright, and inverted, flight, and denoted with lower-case letters (l,r).