Final answer:
The first red-giant phase of a low-mass star is marked by hydrogen shell burning around a contracting core, whereas the second red-giant phase involves helium shell burning around a carbon-oxygen core after the helium flash.
Step-by-step explanation:
Low-mass stars indeed experience two distinct red-giant phases during their evolution. The first red-giant phase occurs after a star exhausts hydrogen in its core and begins to burn hydrogen in a shell around the core. During this phase, the core contracts, the outer layers expand, and the star moves up the red-giant branch on the Hertzsprung-Russell (H-R) diagram, becoming more luminous and cooler at the surface.
The second red-giant phase, also known as the Asymptotic Giant Branch (AGB) phase, occurs after the star has ignited helium in its core, known as the helium flash, and has briefly stabilized while helium burning into carbon and oxygen. Once the helium is exhausted, the star resumes shell hydrogen and helium burning around an inert carbon-oxygen core, becoming a red giant for the second time. This phase is characterized by high luminosity and a cooler surface temperature as the star moves again on the H-R diagram but this time with different internal fusion processes.