'Instability Strip' from fusion PP-to-CNO transition ?

[Widdekind]

fig. 1 --

HR Diagram

 

one-zone approximation for star-core pressures

 

Assuming spherical-symmetry, and hydro-static equilibrium, w.h.t. [math]\frac{dP}{dr} = -g® \rho®[/math]. Then, discretizing that differential equation, and taking solely a single 'step', from star surface to star core (one-zone approximation), w.h.t. [math]-\frac{P_c}{R} \approx - \frac{G M}{R^2} \frac{3 M}{4 \pi R^3}[/math], or (ideal gas law) [math]\frac{\rho_c T_c}{\bar{m}} \approx \frac{3 G}{4 \pi k_B} \frac{M^2}{R^4}[/math].

 

 

 

 

PP [math]\left( 1/2 - 2 M_{\odot} \right)[/math]

 

Observationally (from fig.1), the luminosities, of 'intermediate mass' stars ([math]1/2 - 2 M_{\odot}[/math]), increase with their (photo-spheric) temperatures, approximately as [math]L \propto T^8[/math]. And, since (assuming spherical symmetry) [math]L \propto R^2 \times T^4[/math], that implies that [math]R \propto T^2[/math], for such stars. Meanwhile, (seemingly) separate estimations (mass-luminosity relation) imply, that the luminosities, of such stars, scales as [math]L \propto M^4[/math]. Therefore, in solar-normalized units, w.h.t. [math]\hat{\rho}_c \hat{T}_c \approx \frac{\hat{M}^2}{\hat{R}^4} \approx \frac{\hat{L}^{1/2}}{\hat{L}} \approx \hat{L}^{-1/2} \approx \hat{M}^{-2}[/math] (ignoring effects, of star-core pollution, from fusion products).

 

Next, 'intermediate mass' star fusion is dominated by the proton-proton (PP) process, whose specific power is proportional to [math]T^4[/math]. And, all the power produced by the fusing star-core, is (ultimately) released, at the star-surface, as the star's observed luminosity. Therefore, in solar normalized units, w.h.t. [math]\hat{V}_c \hat{\rho}_c^2 \hat{T}_c^4 \approx \hat{L}[/math]. Now, [math]\hat{V}_c \hat{\rho}_c = \hat{M}_c[/math], so we define a (solar-normalized) 'core mass fraction' s.t. [math]\hat{M}_c \equiv \hat{f}_c \hat{M}[/math]. Then, and dividing through by the mass term, w.h.t. [math]\hat{f}_c \hat{\rho}_c \hat{T}_c^4 \approx \hat{L}^{3/4} \approx \hat{M}^3[/math] (again ignoring effects, of star-core pollution, from fusion products).

 

Then, taking the ratio, of the former to the latter equations, w.h.t. [math]\hat{f}_c \hat{T}_c^3 \approx \hat{L}^{5/4} \approx \hat{M}^5[/math], implying that [math]\hat{T}_c \approx \hat{L}^{5/12} \approx \hat{M}^{5/3}[/math] and [math]\hat{\rho}_c \approx \hat{L}^{- 11/12} \approx \hat{M}^{-11/3}[/math] (assuming the constancy, of the 'core mass fraction'). Approximately, for 'intermediate mass stars', w.h.t.:

 

[math]\hat{L} \approx \hat{M}^4[/math]

 

[math]\hat{T}_c \approx \hat{L}^{1/2}[/math]

 

[math]\hat{\rho}_c \approx \hat{L}^{-1}[/math]

 

[math]\hat{P}_c \approx \hat{L}^{-1/2}[/math]

 

 

 

 

CNO [math]\left( 2 - 20 M_{\odot} \right)[/math]

 

Observationally (from fig.1), the luminosities, of 'high mass' stars ([math]2-20 M_{\odot}[/math]), increase with their (photo-spheric) temperatures, approximately as [math]L \propto T^6[/math]. And, since (assuming spherical symmetry) [math]L \propto R^2 \times T^4[/math], that implies that [math]R \propto T[/math], for such stars. Meanwhile, (seemingly) separate estimations (mass-luminosity relation) imply, that the luminosities, of such stars, scales as [math]L \propto M^{7/2}[/math]. Therefore, in solar-normalized units, w.h.t. [math]\hat{\rho}_c \hat{T}_c \approx \frac{\hat{M}^2}{\hat{R}^4} \approx \frac{\hat{L}^{4/7}}{\hat{L}^{2/3}} \approx \hat{L}^{-2/21} \approx \hat{M}^{-1/3}[/math] (ignoring effects, of star-core pollution, from fusion products).

 

Next, 'high mass' star fusion is dominated by the carbon-nitrogen-oxygen (CNO) process, whose specific power is proportional to [math]T^{17}[/math]. Again, all the power produced by the fusing star-core, is (ultimately) released, at the star-surface, as the star's observed luminosity. Therefore, in solar normalized units, w.h.t. [math]\hat{V}_c \hat{\rho}_c^2 \hat{T}_c^{17} \approx \hat{L}[/math]. Now, again, [math]\hat{V}_c \hat{\rho}_c = \hat{M}_c[/math], so we again define a (solar-normalized) 'core mass fraction' s.t. [math]\hat{M}_c \equiv \hat{f}_c \hat{M}[/math]. Then, again dividing through by the mass term, w.h.t. [math]\hat{f}_c \hat{\rho}_c \hat{T}_c^{17} \approx \hat{L}^{5/7} \approx \hat{M}^{5/2}[/math] (again ignoring effects, of star-core pollution, from fusion products).

 

Then, taking the ratio, of the former to the latter equations, w.h.t. [math]\hat{f}_c \hat{T}_c^{16} \approx \hat{L}^{17/21} \approx \hat{M}^{17/6}[/math], implying that [math]\hat{T}_c \approx \hat{L}^{17/336} \approx \hat{M}^{17/96}[/math] and [math]\hat{\rho}_c \approx \hat{L}^{- 49/336} \approx \hat{M}^{-49/96}[/math] (assuming the constancy, of the 'core mass fraction'). Approximately, for 'high mass stars', w.h.t.:

 

[math]\hat{L} \approx \hat{M}^{3.5}[/math]

 

[math]\hat{T}_c \approx \hat{L}^{0.05}[/math]

 

[math]\hat{\rho}_c \approx \hat{L}^{-0.15}[/math]

 

[math]\hat{P}_c \approx \hat{L}^{-0.10}[/math]

Thus, given the extreme temperature-dependence, of CNO power production, CNO-dominated, 'high-mass' star-cores, operate under quasi-isobaric, quasi-isothermal conditions. So, the cores, of 'high mass stars', are all similar, in bulk properties. And, those star-cores all eventually blow away their outer envelopes, producing Planetary-Nebulae (PNe), cocooning proto-White-Dwarves (proto-WDs):

 

Thus, the similarity, of 'high mass star' pre-WD cores, could account, for the ensuing similarity, of the remnant proto-WDs and WDs, as the enveloping PNe drift away & disperse. Note, too, that the transition, from pre-WDs (T > 50k K), through to isolated WDs (T < 30k K), seems to be associated with the transition from full-to-partial-to-neutral He ionization states (HeIII-to-HeII-to-HeI), and the ensuing decrease in plasma opacity, to interior-generated luminosity (cf. CMBR and Recombination, at z~1100). Does the "characteristic new-born WD surface-temperature layer" (~30-50k K) represent an 'opacity barrier', thermally blanketing the star-core there-within contained ?

 

 

 

 

Unstable Variable Stars [math]\left( \approx 2 M_{\odot} \right)[/math]

 

The transition, from fusion by PP to CNO processes, occurs in the cores of stars slightly more massive than our sun, whose core temperatures exceed ~20M K:

 

And, this transition occurs, for stars near the observed 'break', on the HR diagram, of the stellar luminosity function, at masses of [math]\approx 2 M_{\odot}[/math] (see fig.1). And, this region, on the HR diagram, is also associated with the 'Instability Strip', corresponding to variable stars (e.g. RR Lyrae, Cepheids):

 

Note, that the radial velocities, of the observationally-inferred (effective-)photo-spheric expansions (few AU per day) are characteristic of our sun's "space weather" -- i.e., star-wind (SW), including coronal-mass-ejections (CMEs). Thus, might periodic PP-to-CNO 'power-pulses' may generate rhythmic star-spanning 'super-massive-ejections' ?

 

 

 

 

effects of H depletion ?

 

By definition, the 'average effective mass' [math]\bar{m}[/math], linking total mass-density [math]\rho[/math], to total number-density [math]n[/math], is [math]\frac{1}{\bar{m}} \equiv \frac{X}{m_H} + \frac{Y}{4 m_H} + \frac{Z}{\bar{m}_Z}[/math]. Ignoring the effects, of heavier metal pollutants, upon H-to-He fusion processes ([math]Z \approx 0[/math]), then w.h.t. [math]\frac{m_H}{\bar{m}} \approx X + \frac{Y}{4} \approx \frac{1 + 3 X}{4}[/math]. And, only core H produce power. Thus, accounting for dwindling H mass-fractions ([math]\hat{X}(t) \rightarrow 0[/math]), w.h.t. (solar-normalized units):

 

[math]\hat{\rho}_c \hat{T}_c (1 + 3 \hat{X}) \approx \frac{\hat{M}^2}{\hat{R}^4}[/math]

(HSE)

 

[math]\hat{f}_c \hat{M} \hat{\rho}_c \hat{X}^2 \hat{T}^{4,17} \approx \hat{L}[/math]

(PP,CNO power)

 

[math]\therefore \hat{f}_c \frac{\hat{X}^2}{1+3 \hat{X}} \hat{T}^{3,16} \approx \hat{L} \frac{\hat{R}^4}{\hat{M}^3} \approx \hat{L}^{5/4,17/21}[/math]

So, to maintain constant luminosity, and remain on the Main Sequence (MS), w.h.t.:

 

[math]\hat{T}_c \approx \left( \frac{\hat{X}^2}{1+3 \hat{X}} \right)^{-1/3, -1/16}[/math]

And so, as the star slowly ages, over the eons, [math]\hat{X} \rightarrow 0[/math], the star's core temperature may rise. And, such a hottening, could advantage star-core CNO processes, perhaps accounting for subsequent star swellings, 'up' off the MS, into the Giant Branch (GB) phases ?

 

 

 

 

Qualitatives

 

Externally-observed characteristics (i.e., mass-luminosity relation, on MS) may reflect interior H-fusion processes (e.g., PP, CNO). What, then, about 'very-low mass stars' ([math]< 1/2 M_{\odot}[/math]), and 'very-high mass stars' ([math]> 20 M_{\odot}[/math]) ?

Edited August 6, 2011 by Widdekind

[Widdekind]

"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).

[Widdekind]

"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:

 

Red/Brown Dwarfs - less than 0.6 Ms <== Main Sequence 0.076-0.8 Ms

 

Stars less than about 0.6 solar masses, when nuclear fuel is used up, gravitational collapse shrinks the star, but no more than the gas temperature-pressure-volume laws of classical physics allow. We have not found any white dwarf less massive than 0.6 solar masses. Part of the answer is that the universe may not be old enough for lower mass stars to have evolved off the main sequence (edu-obs).

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) ??

Edited August 10, 2011 by Widdekind

[nyemi]

"fusion PP-to-CNO transition ?"

Sun Panspermia Hypothesis.

Sun----Solar System----- Sun rotation period reduction---- Lithium problem----12C carbon production---Solar flare----Planet -----Carbon based Life Earth

[John Cuthber]

so?

[nyemi]

Hi John

https://en.wikipedia.org/wiki/Standard_solar_model#Lithium_depletion_at_the_solar_surface

7Li=12C

[Enthalpy]

Assuming hydro-static equilibrium, [...]

 

Why? This assumption is abandoned for decades from what I've read. Equilibrium is to result mainly from radiation pressure.