What is the Solar Evaporation Time?

[Orion1]

An interesting equation derived from Classical General Relativity:

Solar evaporation time:
$$\boxed{t_{ev} = \frac{ m_{\odot} c^2}{P_{\odot}}}$$

$$m_{\odot} = 1.98892 \cdot 10^{30} \; \text{kg}$$ - Solar mass
$$P_{\odot} = 3.846 \cdot 10^{26} \; \text{W}$$ - Solar Luminosity

$$\boxed{t_{ev} = 1.473 \cdot 10^{13} \; \text{y}}$$

Understanding a thermodynamic equation similar to this should provide an appreciation for how extraordinarily dynamic and stable our nearest star Sol really is.
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Reference:
http://en.wikipedia.org/wiki/Sun"
http://en.wikipedia.org/wiki/Solar_mass"

 

[Arch2008]

What do you mean by “solar evaporation”? The sun will eventually become a white dwarf which theoretically may not “evaporate” for maybe 10^200 years.

http://en.wikipedia.org/wiki/White_dwarf

 

[Janus]

The problems I see with the expression is that it assumes a constant luminosity for the Sun during its life time and that the Sun will convert all its present mass into energy over its lifetime. Neither of these are true.

 

[Arch2008]

I suppose that he started with Einstein’s equation (E=mc^2) and inserted the definition of a joule for “E”. Then he substituted watt-seconds for the joule and divided both sides by a watt, which gives you something like seconds = mc^2 / watts. Finally he plugged in the Sun’s current mass and luminosity and solved it, which, of course, just doesn’t work.

 

[Helios]

This is very interesting from the point of view of dimensional analysis. I would like to know what Arch means by " just doesn't work ". I note that [ Pressure / Power Density ] also is a certain TIME, but I have no idea what it is or could be called, yet It could be evaluated at the fusion core zone.

 

[Kurdt]

It doesn't work for the reasons Janus pointed out. Neither of the assumptions used are true. There is also the fact the sun loses some small amount of mass through its stellar wind.

 

[Helios]

Kurdt, what meaning did you gather from Arch's post? To me it implies that he finds a flaw in dimensional analysis and has nothing to do with any reasons Janus pointed out. To me it so overtly obvious that such a little equation cannot predict the end of a star's life, that this really needs no comment. Someone just coined the term "solar evaporation time", I doubt that it is loaded with all these assumptions that are being alleged.

 

[Kurdt]

I don't think it implies any flaw with dimensional analysis. Certainly whatever it is has units of time, but Arch's final sentence is the main point of his post for me which relates to what Janus has pointed out.

Why do you doubt its loaded with the assumptions pointed out? If one plugs in the suns mass to E=mc² you assume all that mass is converted to energy etc.

 

[Orion1]

solar wind evaporation...



I attempted a calculation for how much power is evaporated via the solar wind, including rest mass energy evaporation:

Proton plasma kinetic energy temperature and momentum:
$$E_{k} = \frac{3}{2} k_b T_K = \frac{p^2}{2m_p}$$
$$k_b$$ - Boltzmann's constant
$$T_K = 1.6 \cdot 10^6 \; \text{K}$$ - solar wind temperature

Proton plasma momentum and temperature:
$$\boxed{p^2 = 3 m_p k_b T_K}$$

Relativistic kinetic energy:
$$E_k = \sqrt{p^2 c^2 + m^2 c^4} - m c^2$$

Total number of particles carried away from the Sun by the solar wind:
$$\frac{dn}{dt} = 1.3 \cdot 10^{31} \; \text{protons} \cdot \text{s}^{-1}$$

Solar wind evaporation power, including rest mass energy evaporation:
$$\boxed{P_{k} = \frac{dn}{dt} \sqrt{ 3 m_p k_b T_K c^2 + m_p^2 c^4 }}$$

$$\boxed{P_{k} = 1.954 \cdot 10^{21} \; \text{W}}$$
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Reference:
http://en.wikipedia.org/wiki/Solar_wind#Emission"
http://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies"
http://en.wikipedia.org/wiki/Boltzmann_constant#Role_in_the_equipartition_of_energy"

 

[Arch2008]

Helios, if the end product of the dimensional analysis is a solution that is over 180 orders of magnitude off the mark, then that is what I meant by “just doesn’t work”. Orion1’s first link in that post states that the Sun will become a white dwarf, so he should have known this.

So Orion1, what were you attempting with your first post?

BTW, in the second post, when you plugged in the expression for relativistic kinetic energy with the derived value for p into the last equation, why did you drop the “ – mc^2”?

 

[Orion1]

So Orion1, what were you attempting with your first post?

The first post was a routine dimensional analysis with a reality check. I replaced an equation derivative of the model for Hawking radiation black hole evaporation with the solar parameters and produced a solar model and called it the 'solar evaporation time'. This explains why entire evaporation of the model was assumed by the equation and why the evaporation rate is linear and why the equation still requires an integration for mass and the dimensional area under the power dissipation function as a function of mass or time.

Hawking radiation black hole evaporation time derivative equation:
$$\boxed{dt_{ev} = \frac{ dm_{bh} c^2}{P_{bh}(m)}}$$

Your observational answers suggest to me that Hawking radiation black hole evaporation is an existential observation in a free space model, without any prior knowledge if its interior dynamics with respect to time. This only suggest to me that black holes may have a dynamic quantum energy interior that could effect its total dynamics, such as evaporation lifetime.

when you plugged in the expression for relativistic kinetic energy with the derived value for p into the last equation, why did you drop the “ – mc^2”?

Because the sun evaporates particle rest mass and kinetic energy. The evaporated particle rest mass was actually added to the kinetic energy equation to account for the total loss of evaporated particles rest mass and its kinetic energy, which canceled the existing rest mass term in the relativistic kinetic energy equation.
$$E_t = E_m + E_k$$

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