Cooling of Planetary Nebula pre-White Dwarves

[Widdekind]

Planetary Nebula (PNe) consist of an increasingly exposed "pre-WD" star core, surrounded by the increasingly distant, diffuse, and (often) ionized shed outer layers, of said star. After the failure of He fusion, the post-fusion, "pre-WD" star core remnant quickly cools down, from ~100k K, to ~30k K, in ~10 kyr:

 

[math]L = 4 \pi R^2 \times \sigma T^4 = -\frac{dE}{dt}[/math]

 

[math]E \approx N k_B T \approx \frac{M}{m_H} k_B T[/math]

 

[math]\therefore \frac{dT}{dt} \approx -\frac{4 \pi R^2 \, \sigma}{N k_B}T^4[/math]

 

[math]\therefore \frac{1}{T_i^3} - \frac{1}{T_f^3} \approx \frac{4 \pi R^2 \, \sigma}{3 N k_B} \times t[/math]

Plugging in characteristic values ([math]T_i \approx 100k \, K[/math], [math]T_f \approx 30k \, K[/math], [math]R \approx 0.01 R_{\odot}[/math], [math]M \approx M_{\odot}[/math]), I compute a 'cooling time scale' of [math]\tau \approx 2 Myr[/math]. This is more than two orders-of-magnitude more, than the observed characteristic cooling time, of [math]\approx 10 Kyr[/math]. This discrepancy could be simply explained, by assuming that only the outer layers of the pre-White Dwarves actually cool down. Er go, their interiors remain a high-temperature, high-pressure mixture of carbon & oxygen, or "molten diamond".

 

Fig. 1 --

stars of [math]\approx 1-8 M_{\odot}[/math] blow off outer layers, gradually generating PNe, surrounding the increasingly exposed star-cores, down to ever deeper once-interior depths. Mass-loss may cease, along with He fusion in the spent star-core, by the time that the now-exposed 'surface' layer, of the central pre-WD, reveals a temperature of [math]\approx 100k K[/math]. As the pre-WD cools down, to a temperature of [math]\approx 30k K[/math] (

i.e.

, [math]\approx 13.6 eV[/math]), its harsh-and-ionizing radiation produces observed PNe. Afterwards, the PN fades, 'decoupling' from the now-comparatively cold WD, which appears as an independent astronomical object.

 

 

Missing Mass in WD-PN systems ?

 

PNe contain [math]\approx 0.1-1 M_{\odot}[/math] of observed circum-stellar plasmas. And, PNe are apparently generated, by progenitor-stars, of [math]\approx 1-8 M_{\odot}[/math]. But, WD remnants are predicted to mass only [math]\approx 1.4 M_{\odot}[/math]. This leaves up to [math]\approx 6.5 M_{\odot}[/math] of progenitor-star material unaccounted for.

Edited July 31, 2011 by Widdekind

[Realitycheck]

If a star has progressed in evolution to its carbon stage, what prompts you to state that the carbon form produced is diamond? Why not graphene or buckeyballs or just graphite? Seems like there would be a lot more diamonds in existence if they were produced in stars.

Edited August 1, 2011 by Realitycheck

[Widdekind]

Purely putting it poetically