M-dwarf stars (red dwarfs) are the most common stellar targets in the search for habitable worlds. Astronomers are particularly interested in detecting "Super-Earth" planets (with masses Mp≈5M⊕ M_p \approx 5 M_\oplus\,Mp≈5M⊕ and radii Rp≈1.5R⊕R_p \approx 1.5 R_\oplusRp≈1.5R⊕) orbiting within the close-in habitable zones of these cool stars, typically at an orbital radius of 0.1 AU0.1\text{ AU}0.1 AU.
Discuss the physical principles and the specific observational advantages or challenges associated with using the following three methods to detect such a planet orbiting a typical M-dwarf star (Ms≈0.3M⊙M_s \approx 0.3 M_\odotMs≈0.3M⊙, Rs≈0.3R⊙R_s \approx 0.3 R_\odotRs≈0.3R⊙):
In your answer, compare the feasibility of these methods for this system compared to an Earth-Sun twin system, and justify which of the methods has been most successful for confirming planets in these compact M-dwarf systems.
