# Qual o tamanho de uma bola de água sem iniciar a fusão?

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Qual o tamanho de uma bola de água sem iniciar a fusão?

Pergunta peculiar: alguma explicação pode ser necessária. Meu filho está no 'espaço' e na astronomia. Um de seus cartazes diz que Saturno poderia flutuar, se um oceano suficientemente grande pudesse ser encontrado. Obviamente, isso não funcionaria: a atmosfera de Saturno se desprenderia e se juntaria ou se tornaria a atmosfera do corpo maior, e então o núcleo denso de Saturno afundaria.

Mas esse oceano poderia existir sem a fusão começar?

Why is there an assumption that this ocean is a big ball of water? Surely it is pictured being a vast puddle on an even vaster plain on a vast hollow planet? Then there would be no fusion. IE I don't think that just because a ball of water can't be created that is big enough means that the proposition itself is fundamentally unsound.

Why the need to ask "why", @GreenAsJade? The OP has painted the scenario of Saturn swimming in the ocean of a vastly larger "planet", so let's roll with that. It's not about Saturn, but about the planet (a.k.a. a sun-sized body/blob of water).
AnoE

Marginally related: what-if.xkcd.com/4 , "a mole of moles"
Carl Witthoft

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@AnoE The reason I asked why is because the answers conclude that Saturn could not float in an ocean of water, based on the assumption that the ocean we speak of is a large spherical blob of water that would fuse. However, the "childrens' story" that "Saturn would float" is not based on such an assumption. If you are going to get all sciency-pedantic about a story for children who's purpose is simply to make them think about what density means, then you need to be sciency-pedantic about the assumptions. The OP assumed the ocean is a blob of water, but no real ocean is a blob.

@GreenAsJade Essa é uma resposta justa. A água precisa ser quase tão profunda quanto o diâmetro de Saturno. Se estivesse em um planeta oco muito grande (detalhes de engenharia, TBD), isso poderia funcionar? Haveria problemas com a quantidade "horizontal" de água, estendendo-se até o horizonte por vários diâmetros de Saturno? Isso implicaria vários volumes de água de Saturno nas proximidades: voltamos às conseqüências da gravidade?
Jdaw1

Respostas:

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Você realmente precisa de um modelo de evolução estelar completo para responder a isso com precisão e não tenho certeza se alguém faria isso com uma estrela dominada por oxigênio.

To zeroth order the answer will be the similar to a metal-rich star - i.e. about 0.075 times the mass of the Sun. Any less than this and the brown dwarf (for that is what we call a star that never gets hot enough at its centre to initiate significant fusion) can be supported by electron degeneracy pressure.

A star/brown dwarf with the composition you suggest would be a different. The composition would be thoroughly and homogeneously mixed by convection. Note that other than a thin layer near the surface, the water would be completely dissociated and the hydrogen and oxygen atoms completely ionised. Hence the density of protons in the core would be lower for the same mass density than in a "normal star". However, the temperature dependence is so steep I think this would be a minor factor and nuclear fusion would be significant at a similar temperature.

Of much greater importance is that there would be fewer electrons and fewer particles at the same density. This decreases both the electron degeneracy pressure and normal gas pressure at a given mass density. The star is therefore able to contract to much smaller radii before degeneracy pressure becomes important and can thus reach higher temperatures for the same mass as a result.

For that reason I think that the minimum mass for hydrogen fusion of a "water star" would be smaller than for a star made mainly of hydrogen.

But how much smaller? Back of the envelop time!

$\mathrm{\Omega }$$\Omega$

$P=\rho kT/\mu {m}_{u}$$P = \rho kT/\mu m_u$, where $T$$T$ is the temperature, $\rho$$\rho$ the mass density, ${m}_{u}$$m_u$ an atomic mass unit and $\mu$$\mu$ the average number of mass units per particle in the gas.

Assuming a constant density star (back of the envelope) then $dV=dM/\rho$$dV = dM/\rho$, where $dM$$dM$ is a mass shell and $\mathrm{\Omega }=-3G{M}^{2}/5R$$\Omega = -3GM^2/5R$, where $R$$R$ is the "stellar" radius. Thus

$\frac{G{M}^{2}}{5R}=\frac{kT}{\mu {m}_{u}}\int dM$
$T=\frac{GM\mu {m}_{u}}{5kR}$
and so the central temperature $T\propto \mu M{R}^{-1}$$T \propto \mu MR^{-1}$.

Now what we do is say that the star contracts until at this temperature, the phase space occupied by its electrons is $\sim {h}^{3}$$\sim h^3$ and electron degeneracy becomes important.

A standard treatment of this is to say that the physical volume occupied by an electron is $1/{n}_{e}$$1/n_e$, where ${n}_{e}$$n_e$ is the electron number density and that the momentum volume occupied is $\sim \left(6{m}_{e}kT{\right)}^{3/2}$$\sim (6m_e kT)^{3/2}$. The electron number density is related to the mass density by ${n}_{e}=\rho /{\mu }_{e}{m}_{u}$$n_e = \rho /\mu_e m_u$, where ${\mu }_{e}$$\mu_e$ is the number of mass units per electron. For ionised hydrogen ${\mu }_{e}=1$$\mu_e=1$, but for oxygen ${\mu }_{e}=2$$\mu_e=2$ (all the gas would be ionised near the temperatures for nuclear fusion). The average density $\rho =3M/4\pi {R}^{3}$$\rho = 3M/4\pi R^3$.

Putting these things together we get

${h}^{3}=\frac{\left(6{m}_{e}kT{\right)}^{3/2}}{{n}_{e}}=\frac{4\pi {\mu }_{e}}{3}{\left(\frac{6\mu }{5}\right)}^{3/2}\left(G{m}_{e}R{\right)}^{3/2}{m}_{u}^{5/2}{M}^{1/2}$

Thus the radius to which the star contracts in order for degeneracy pressure to be important is
$R\propto {\mu }_{e}^{-2/3}{\mu }^{-1}{M}^{-1/3}$

If we now substitute this into the expression for central temperature, we find

$T\propto \mu M{\mu }_{e}^{2/3}\mu {M}^{1/3}\propto {\mu }^{2}{\mu }_{e}^{2/3}{M}^{4/3}$

Finally, if we argue that the temperature for fusion is the same in a "normal" star and our "water star", then the mass at which fusion will occur is given by the proportionality

$M\propto {\mu }^{-3/2}{\mu }_{e}^{-1/2}$
.

For a normal star with a hydrogen/helium mass ratio of 75:25, then $\mu \simeq 16/27$$\mu \simeq 16/27$ and ${\mu }_{e}\simeq 8/7$$\mu_e \simeq 8/7$. For a "water star", $\mu =18/11$$\mu = 18/11$ and ${\mu }_{e}=9/5$$\mu_e= 9/5$. Thus if the former set of parameters leads to a minimum mass for fusion of $0.075{M}_{\odot }$$0.075 M_{\odot}$, then by increasing $\mu$$\mu$ and ${\mu }_{e}$$\mu_e$ this becomes smaller by the appropriate factor $\left(18×27/11×16{\right)}^{-3/2}\left(9×7/5×8{\right)}^{-1/2}=0.173$$(18\times 27/11\times 16)^{-3/2} (9\times 7/5\times 8)^{-1/2} = 0.173$.

Thus a water star would undergo H fusion at $0.013{M}_{\odot }$$0.013 M_{\odot}$ or about 13 times the mass of Jupiter!

NB This only deals with hydrogen fusion. The small amount of deuterium would fuse at lower temperatures. A similar analysis would give a minimum mass for this to occur of about 3 Jupiter masses.

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A splendid analysis of a water star, much of which was beyond my expertise. But 13 M♃ is sufficiently small that its radius would be about thrice that of Saturn, far too small for Saturn even to try floating — ignoring the minor practical issues. So the comment on my son’s poster, which I recall being used in my long-lost youth, is really stupid. Thank you.
jdaw1

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@jdaw1 Water is not present at several million degrees...
Rob Jeffries

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@KRyan edit made so that is now crystal clear. There is H and O - completely ionised and thoroughly mixed.
Rob Jeffries

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@jdaw1 Water is very much compressible at the pressure inside a planet or gas giant. I just want to add, the chemistry would make a "water world" kind of impossible long before 12 or 13 Jupiter masses. The chemistry inside the planet would likely split the water molecules and you'd have a hydrogen atmosphere gas giant that looks nothing like a water world at 1 Jupiter mass, probably even less. The practical limit to a water world that looks like a water world is probably lighter and smaller than Saturn.
userLTK

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@Aron perhaps you could explain what you mean? There is no "experimental evidence" on the issue. The star you mention is a white dwarf., supported by electron degeneracy pressure and containing almost no hydrogen. The temperature for oxygen fusion is much higher than H fusion by a factor of >500. Factoring that in, my back-of-envelope calculation suggests a minimum mass for O fusion of about 0.7 solar masses. A correct stellar evolution calculation would show that a C/O core needs to grow to just over 1 solar mass to begin fusion. I'll accept that level of accuracy.
Rob Jeffries
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