What would be the diameter of a sphere of iron with the mass of our sun? Use standard density of metalic iron. I realize it would then collapse into a star in its own right and become much smaller as it’s density increases. How does this relate to the large diameter that our sun will have when at its red dwarf stage? Thank you.
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The object would shrink to become an "iron white dwarf", supported by electron degeneracy pressure.
If you want to assume that you have a cold sphere of iron supported by electron degeneracy pressure, then you can use the Nauenberg (1972) approximation for the radius of a cold$^*$ white dwarf with $\mu_e$ mass units per electron and a Chandrasekhar mass $M_C = 5.816 M_{\odot}/\mu_e^2$. $$ R = \frac{2.45354}{\mu_e} R_{\rm Earth} \left(\frac{M}{M_{C}}\right)^{-1/3} \left[ 1 - \left(\frac{M}{M_c}\right)^{4/3}\right]^{1/2}$$
Thus for $^{56}$Fe, $\mu_e = 2.15$, $M_c = 1.26M_{\odot}$, and if the mass of your star is $M=1M_{\odot}$, we have $R= 0.635R_{\rm Earth} \simeq 4045$ km.
The Sun will end its life as a white dwarf with a mass of about $0.5M_{\odot}$ and will be made of a mixture of carbon and oxygen with $\mu_e =2$ and $M_c = 1.45M_{\odot}$. Putting this into the Nauenberg approximation we have an estimated radius for the end-of-life Sun as a cold white dwarf of $R = 9700$ km.
$^* $ "Cold" in this context means that the Fermi kinetic energy is much greater than $k_B T$. In practice this means $T<10^9$ K. A contracting iron ball roughly in hydrostatic equilibrium will never reach such temperatures because of highly efficient neutrino emission above $10^8$ K.
ProfRob
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Hint: I'm assuming you know (or can find) the mass of the Sun $m_\odot$ and the density of solid iron on Earth $\rho_{Fe}$. Now, you just need to know the relation between density, mass and volume.
Regarding what happens after someone created such a sphere, I don't know for sure. The mass is below the Chandrasekhar limit of $1.4 M_\odot$, so it will not become a neutron star. (If it was the case, the diameter would be significantly reduced because its density would become around $10^{14}$ larger than the density of the Sun.) Nevertheless, the extra pressure due to the mass would decrease the diameter of the iron sphere since it's density would rougly be $10^6$ the density of the Sun.
If it had enough energy (e.g, from approaching the iron to create a sphere), we may get a white dwarf. In time it would cool down and it's colour would go to yellow, red, and brown, since the colour depends on the surface temperature (hotter means whiter and brighter). However, red dwarfs and brown dwarfs are different types of object.
