The first stage of stellar evolution after hydrogen fusion begins in a star's core is called the main sequence. The moment fusion starts, the star is zero age main sequence (ZAMS). The moment hydrogen runs out in the core of the star is its terminal age main sequence (TAMS). The star then moves on to the next stage.
The ZAMS mass of a star doesn't completely determine its evolution. Metalicity, the amount of elements besides hydrogen and helium, can effect the evolution through e.g. extra mass loss from stellar wind. Rotation also has effects. The below ZAMS masses are therefore approximate.
Metalicity is usually denoted by $Z$, where $X$ is the stellar mass fraction of hydrogen, $Y$ is the stellar mass fraction of helium, and $Z$ is the remaining stellar mass fraction. So, $X+Y+Z = 1$.
From Iben 1985
Intermediate mass stars burn hydrogen in their cores on the main sequence until the core becomes mostly helium. Core fusion ceases, the core shrinks, and fusion begins in a thick shell of hydrogen around the core. This is the sub-giant stage. The heat generated in this stage pushes outer layers outward, thinning the fusion shell and expanding the star until it reaches the red giant stage.
The red giant grows larger and brighter, while the surface cools. During this time, more hydrogen is converted to helium, and the helium core increases in pressure and temperature until helium fusion begins. The star's surface then cools and shrinks and the star moves onto the horizontal branch. This is the core helium burning stage. Notably, helium burning doesn't contribute much to the overall energy generation of the star; shell hydrogen burning dominates energy generation.
Once the core runs out of helium, the star moves onto the asymptotic red giant branch (AGB), where a carbon core is surrounded by a helium burning shell, surrounded by a hydrogen burning shell. Much of the outermost hydrogen layer can be lost at this stage, through stellar wind, thermal pulses, and, finally, a period of super wind. This super wind envelope is what later becomes a planetary nebula (PN). When helium is exhausted, and hydrogen is exhausted or lost, the star shrinks and increases in temperature until it eventually is hot enough to ionize the envelope into a PN. The star then turns into a white dwarf.
From Iben 1967
Stars with ZAMS mass < 0.5 M☉ stay on the main sequence longer than the age of the universe, so no observations of evolved states are possible. They are important, though, because lower mass stars are more common in the universe. On the main sequence, these stars are fully convective, so a thorough mixing of hydrogen fuel and helium happens.
Stars with ZAMS mass more than 0.18 M☉ do eventually move off the main sequence into the red giant stage. These stars aren't massive enough to start helium fusion, so become white dwarfs after burning, or expelling, the remainder of their hydrogen.
Stars with ZAMS mass less than 0.18 M☉, don't have enough mass for hydrogen shell burning around a helium core, so go directly from the main sequence to become white dwarfs composed of helium.
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