Evolution of single stars

Evolution of single stars [@TaurisvandenHeuvel+2023]

← Physics of Binary Star Evolution

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Why stars do stuff (trying to focus on some of the math here, cause while i conceptually understand it, the math is really neat)

• A globe of monatomic gas without energy sources and in HSEq follows
• $2E_{th}+E_{pot}=0$
• $E_{th}$ is given by
• $E_{th}=\frac{3}{2}Nk\bar{T}=\frac{3}{2}M(\mathcal{R}/\mu )\bar{T}$
• Where $N$ is the partcile number in the star, $k$ is the boltzman constant, $M$ is the mass of the globe $\mathcal{R}=ak/m_h$, the ideal gas constant, $\mu$ is the mean particle mass, in units of $m_h$ of the hydrogen atom
• $E_{pot}$ is given by
• $E_{pot}=-\alpha G{M}^2/R$
• Where $R$ is stellar rad, $G$ is grav const, and $\alpha$ is a constant of proportionality of order unity, which depends on the density distribution of the star.
• From substitution, we find that
• $\bar{T}=\alpha (\frac{GM}{R})(\frac{\mu }{3\mathcal{R}})$
• This is import because it shows that internal temp is only depended on the stellar radius, increasing when the star shrinks
• Energy loss is given by
• $E_{tot=E_{th}+E_{pot}}=\frac{1}{2}{E}_{pot}=-\alpha \frac{G{M}^2}{2R}$
• This shows that as $E_{tot}$ decreases the radius of the star must decrease
• However, as shown by $\bar{T}$,as the star contracts the internal temp increases
• This means as the star (or cloud of gas) radius heat away, it actually gets hotter, leading to more radiation, and thus more shrinking
• This applies to the star from the moment it is a gas to the end of its life as BH, NS, WD, etc
• These equations work well for antibiotic-index of $\gamma =C_p/C_V=5/3$, which is great for globes of ionized hydrogen and helium. However, generalized forms can be found with eqs 8.6-8.8
• if $\gamma \le 4/3$, the star cannot reach HSEq, and thus must collapse or explode
• Stars of very high mass have very high luminosities, which mean their interior pressure is dominated by photon-gas, which has $\gamma =4/3$. This sets an upper limit for the mass of a star, also called the Eddington Luminosity Limit

Stellar Timescales

There are three timescales for single star evo that are relevant for binary stellar evo

Dynamical Pulsation timescale

Dynamical-Pulsation-timescale

${\tau }_{dyn}=\frac{R}{c_s}\sim eq50\text{min}{\left(\frac{\overline{{\rho }_{\odot }}}{\bar{\rho }}\right)}^{1/2}$ Where $\bar{\rho }$ is the mean mass density.

This is the timescale of how long it takes for a start to restore a perturbation of its HSEq. This can be defined as the time it takes for a sound way with velocity $c_s$ to cross the stellar radius

Thermal/Kelvin-Helmholtz timescale

Timescale of how long it takes for the star to react to fusion rate not being equal to the radiative energy loss. This is import with pre-main sequence contraction and after the stars fuel has been used

Nuclear timescale

Time it takes for a star to use all of its available fuel

High mass evolution$M\ge 12M_{\odot }$

• Leave behind a collapsing iron core, which creates a NS or BH

Low mass stellar evolution $M\le 8M_{\odot }$

• The degenerate mass in the core of the star heavily effects fusion
• For electron degenerate gas, the pressure only depends on the density (and not on the temperature)
• This means that this degenerate gas ignites, it has no way of stabilizing itself, leading to a `flash’, where it all ignites rapidly.
• This will only stop when the temp reaches a point where the ideal gas is able to also do fusion, at which point the star can actually expand and cool
• “In stars with $M<2.3M_{\odot }$, the core becomes degenerate during hydrogen shell burning, and when $M_{he}\approx .47M_{\odot }$, the helium ignites with a flash, the temp rises to $\approx 10^9$K, and the degeneracy is removed”
• This is not violent to actually disrupt the star
• In stars with mass $2.3m_{\odot }<M\lesssim 8M_{\odot }$ they instead ignite carbon in a flash. This is strong enough to disrupt the star (albeit rarely)
• However, it is more likely for the star to eject its helium envelope due to helium-shell burning as well as the instability of the RSG stage, leaving behind a CO WD.
• Because of this CO ignition is rare.

Mass limit at $\sim 1.2M_{\odot }$

• When hydrogen is exhausted in the star, the star contracts. This causes it to drift sharply left on the HR diagram, until the hydrogen-shell begins fusion causing it to have drift slowly upward and to the right on an HR diagram

Mass limit at $\sim 1.5M_{\odot }$

• Masses less than $\sim 1.5M_{\odot }$ have convective outer envelope and ones higher are radiative.
• This convective envelope creates a magnetic field, this magnetic field can cause Magnetic Breaking, leading to stars of this mass range having slower spins

Stars in the range of $8-12M_{\odot }$

• Not very well is known about evolution in this range
• Generally, the carbon in the ore will ignite and leave a degenerate ONeMg core
• This happens after they eject their hydrogen envelope, but in binaries this envelope is lost through mass transfer
• This means that the ONeMg core will grow to the Chandrasekhar Limit, at which point it will then collapse, creating NS and SN explosion
• Might also result in TI-SNe

Effects of wind mass loss, metallicity, and rotation

• If a star has very fast spin, the helium in the core can get mixed into the whole star, preventing the star from becoming a giant, instead leading it towards becoming a WR-star (This is cool as shit. Blender star my beloved). This can happen with stars with of low of mass as $15M_{\odot }$, as compared to the typical progenitor mass of $\sim 25M_{\odot }$
• Non-rotating stars can become much more massive %
• RSGs are much more common which stars of higher (sun-like) metallicities

Final Evo of stars in the range of $1-8M_{\odot }$

• Unstable pulsing
• Very strong stellar winds
• If they’re low enough mass, ($<8M\odot$), they can become WDs before carbon ignition

Final Evolution and core collapse of stars more massive than $8M\odot$

Between 8 and $\sim 10-12$

• When the core approaches the Chandrasekhar Limit thus begins the onset of core collapse