The goal of this question was for e to provide a proof that the Absolute Peak Luminosity of type 1A Supernovae have a G^(-3) dependence.
The argument is correct but it seems to be too complex.
There is a simpler argument that people can understand better. Just follow these links.
https://www.researchgate.net/post/Does_the_Supernova_Apparent_Distance_scales_with_G3_2
https://hypergeometricaluniverse.quora.com/Second-Peer-Review-2
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Supernovae distances are mapped to the Absolute Peak Luminosity of their light profiles. This means the the only two measured values are luminosity at the peak and and 15 days later (to measure width).
Supernova explodes through a nuclear chain reaction:
1) C+C->Mg
2) Mg+O->Ca
3) Ca+O->Ni
4) Ni->Co->Fe
Luminosity is equal to the number of Ni atoms decay per second or dNidt.
So the peak Luminosity is the Peak dNidt.
There are TWO considerations that together support my approximation:
a) The detonation process accelerates 2-3 reactions (in comparison with equilibrium rates prior to detonation).
b) The detonation process adds a delay to photon diffusion. The shock wave originated in the core will travel to the surface. When the shock wave arrives at the surface, reaction 1-3 should (in principle) stop. Ejecta (non burned residues) are then eject and the photons resulting from the Ni decay have to diffuse through the thick ejecta cloud.
If you look into the Light/[C]^2 curve, you will realize that is has a small delay with respect to Light curve. The constraint of having a finite star size forces the maximum absolute peak luminosity to synchronize itself with the maximum peak Magnesium rate dMgdt, which happens at the maximum radius. So, the Physics of a finite star and a shockwave nuclear chemistry process forces the Peak Absolute Luminosity (dNidt) to match the maximum rate of Magnesium formation (dMgdt). Implicit in this conclusion is the idea that the pressure and temperature jump expedites intermediate fusions.
My contention is:
a) Light has to go through a diffusion process while traveling from the core. The motion of the detonation curve might synchronize light and [C]^2
b) The model in the python script contains a parameter associated with the light diffusional process leading to the peak luminosity.
I would love to hear about the chosen rate values (I used arbitrary values that would provide a time profile in the order of the observed ones). I would appreciate if you had better values or a model for rewriting the equations for the nuclear chain reaction.
I see the detonation process as a Mg shock wave propagating through the star. Light would follow that layer and thus be automatically synchronized with [C]^2
Under these circumstances, volumetric nuclear chemistry depicted in the python script would have to be replaced by shockwave chemistry. That would certainly be only dependent upon the Mg content on the shockwave and thus make light be directly proportional to [C]^2!!!
In Summary:
HU see the Supernova Light process to be proportional to [C]^2. This assertion has support on two mechanisms:
Since I wrote this, I followed up on my own suggestion and considered the shockwave nuclear chemistry approach. You can download all my scripts at the github below.
The shockwave model considers that the amount of light on a cell along the shockwave is is the integrated light created through its evolution. It is developed as a unidimensional process since the observation (billions of years away from the supernova) can be construed as having only contributions from all the cells along the radial line connecting us to the Supernova.
So, the model is unidimensional. That said, it contains all the physics of a tri-dimensional simple model. All rates are effective rates since during the Supernova explosion nuclear reactions are abundant (one can have tremendous variations on neutron content).
The physics is the following:
a) White Dwarf reaches epoch-dependent Chandrasekhar mass. Compression triggers Carbon detonation A shockwave starts at the center of the White Dwarf
b) That shockwave induces 2C->Mg step. The energy released increases local temperature and drive second and third equation to the formation of Ni. Ni decay releases photons.
c) Photons follow the shockwave and diffuse to the surface where we can detect them. The shockwave takes tc to reach the Chandrasekhar radius (surface of the White Dwarf).
d) Luminosity comes from the Ni decay from the element of volume plus the aggregate photons traveling with the shockwave. They diffuse to the surface
e) Two diffusion rates are considered. One for light diffusion within the Star and another for diffusion in the ejecta.
# Diffusion process with two rates 0.3 for radiation created before the shockwave
# reaches surface and 0.03 for radiation diffusion across ejecta
kdiff=0.3*(ttc)
f) I considered tc to be 15 days, that is, it takes 15 days for peak luminosity. Changing this value doesn't change the picture.
g) The peak luminosity is matched to the peak Magnesium formation at t=tc or when the shockwave reaches the Star surface.
This means that Physics makes the Absolute Luminosity Peak to be also the peak of Magnesium formation and that takes place at the Star surface.
https://issuu.com/marcopereira11/docs/huarticle
The mechanism should agree with the observed ratios of ejected mass and the remnant star.
Read the Nuclear Chemistry Reactions. I didn't change them, so as long as the chain reaction is described by the equations I used, the the products will follow. There is the caveat that one cannot use steady state rates. There is a temperature/pressure jump traveling with the shockwave.
# dcarbondt=-kmg*[C][C]
dcarbdt=-kmg*carbon0**2
# dMgdt=kmg*[C][C]-kca*[Mg][O]
dmgdt=kmg*carbon0**2-kca*mg0*ox0
# dCadt=kca*[Mg]*[O]-kni*[Ca][0]
dcadt=kca*mg0*ox0-kni*ca0*ox0
#doxdt=-kca*mg0*ox0-kni*ca0*ox0
doxdt=-kca*mg0*ox0-kni*ca0*ox0
# dNidt=kni*[Ca][O]-klight*[Ni]
# We consider Calcium + Oxigen fusion to take place only until the shockwave reaches surface
# knidecay refers to the decay Ni->Co->Fe
dnidt=kni*ca0*ox0*(t
Another argument to map Absolute Peak Luminosity (APL) to G^(3)
This is what I want to prove (or learn that it is unprovable) APL proportional to G^(-3)!!!!
This is equivalent to say that APL proportional to [C]^2 if [C] proportional to G^(-3/2).
Here goes another argument.
We know that the total mass will be G^(-3/2) due to Chandrasekhar mass G dependence. The time integrated luminosity should scale with Mass squared since Carbon is proportional to mass....This statement considers that the intermediate 40Ca, 24Mg will mostly be consumed to form 56Ni.
It seems to be an appropriate assumption. If not completely consumed, at least the fraction of Ni created is independent upon G (which is supported by the independence of density with G)
The assumption is that the integral of total time profile is being calculated and considered to be conserved by WLR. WLR is an empirical adjustment, so there is no mentioning of integration. This is consistent with a this WLR statement: a wider light profile maps to a higher absolute light peak, because in normalizing the width, one has to increase the APL to keep the same area under the profile.
If one normalizes all time profiles through WLR, we are normalizing the integrated energy and that is proportional to G^(-3) which imply what I want to conclude.
Let me know which argument pass and if you can find a failure on the others.
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This is necessarily an imperfect argument due to the inclusion of an empirical adjustment on the measurement.
I am not familiar enough with the community to know if they believe WLR does Light profile reshaping while keeping energy conservation. It is obvious, but I know that obvious things pass over the community's head every other day..:)
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I’m also skeptical that the luminosity of a SN Ia if G were different would scale as G^-3 (or M_ch^2).
Ni-56 production is not a simple rate-limited process; SNe Ia undergo a deflagration that (in most cases) transitions to a detonation. They burn about half their mass to Ni-56 (depending on when the detonation occurs). Even if Ni-56 production were a simple process, the radius (and thus the density) of the white dwarf also changes with G.
This is the relevant part of the review from one of my ScienceDirect Reviewers.
It is basically just a mere opinion, not based on facts..:) Offered no model to prove me wrong and it included an incorrect fact. He mentioned that density would change, displaying that he/she was not only ignorant of the subject but also didn't read my Appendix where the derivation of Chandrasekhar density is calculated.
Ignorant of the subject Supernova is an unlikely event. The person is ignorant of this perspective (G-dependent SNe). I sent a rebuttal but never heard back.
The radius has a G^(-1/2), the Mass has a G^(-3/2).. the density has no G dependence.
So this person missed the very important aspect of G-Dependence, that is, the pressure profile doesn't change. Only the radius or volume changes.
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I received in personal emails this statement: It is wrong!. I am still trying to find out if my derivation is wrong or if the conclusion is wrong. Those are two different things.
Derivation involves some mental gymnastics due to introduction of empirical methodology in the calculation of distances. There is nothing I can do. The cleanest reasoning is this simplest one, which I've just presented.
I struggled with relating the peak luminosity directly with G. This logical path has many obstacles - assumption of coasting velocity, WLR, not to mention people asking me to make more and more complex models for the detonation itself.
So, it seems to be clear that I should follow the straight path. This question about [C]^2 dependence is just to lead the reader to the right conclusion. The conclusion is right. The argument is confusing. So I will end this effort to lead you to my conclusion using the nuclear reactions.
Check the straight dope. No BS ... just the facts, below.
https://www.researchgate.net/post/Does_the_Supernova_Apparent_Distance_scales_with_G-3_2
https://www.researchgate.net/post/Does_the_Supernova_Apparent_Distance_scales_with_G-3_2
@Jerry Decker
I will delete this question soon because it became a lesser interesting path to prove what I 'proved' here.
Please, feel free to correct my reasoning.
https://www.researchgate.net/post/Does_the_Supernova_Apparent_Distance_scales_with_G3_2
Researchgate doesn't allow for deletion of a question, so I will just replicate a derivation of Luminosity proportional to G^(-3).
Dr. David Arnett was generous to point out that he didn't agree with my argument using the thermonuclear chain reactions. It is not clear that he disagreed with the final conclusion.
In any event, Dr. Arnett made an imprint on me. Like a duckling recently hashed, Dr. Arnett's work was the first one I found modeling Luminosity. So, he is like a (duckling) mother to me...:)
If, my idea were to be correct, it would probably have to be correct within the logical framework created by Dr Arnett's oeuvre.
So, I searched his work for equations of Luminosity of Supernovae. Found one from 1980. I started with equation (40) and revied what influence a variable G would have on it. Since the radius of this Supernova varies according to Chandrasekhar radius or G^(-1/2), and since epoch-dependent Supernovae are scaled down Supernovae, their thermal energy (a volumetric integral) would scaled with G^(-3/2)... Left was the calculation of the mass of the Sun within that context.
I understood as Mass of the Sun, not as the mass of our current Star Sun, but as a unit of mass (currently matching the Sun's mass) but in the context of a Supernova.
The context of a Supernova means that radiative pressure outweighs gas pressure. In that regimen, all the Gravitational dependence of the Sun's mass is included in its mass.
L (Sun) is proportional to Sun's mass.
The Luminosity of the star is also directly proportional to its surface area. That bring about another G(-1). The last G(-1/2) comes directly from a R factor. So:
Thermal energy scales with G(-3/2)
Sun Mass has a dependence of G^(-1)
Radius scales with G^(-1/2).
The total dependence of the Supernova Luminosity is the product of all these dependences or G^(-3).
So, I obtained the same result from my simple-minded physical reasoning using the infinitely more sophisticated work of Dr. David Arnett.
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