Astrophysics of planets orbiting stars other than the Sun. Detection methods: (i) transit — photometric dip in the host star's flux as the planet passes the stellar disc, depth δ = (R_p/R_s)² to leading order (Mandel-Agol 2002 transit…
exoplanets
Transit geometry: dip depth δ = (R_p/R_s)²; duration ∝ R_s/v_orb
Transit photometry detects an exoplanet via the periodic dip in the host star's flux as the planet crosses the stellar disc. Geometry: (i)…
Radial-velocity K ∝ m_p sin i / M_s^{2/3}: stellar Doppler wobble
Radial-velocity (RV) method detects an exoplanet via the Doppler shift of stellar absorption lines as the host star wobbles around the…
T_eq = T_s · √(R_s/(2a)) · (1−A)^{1/4}: blackbody energy balance
Equilibrium temperature of a planet — the temperature at which an idealised blackbody re-emits the absorbed stellar flux in thermal…
Transit depth R_p/R_s = 1/10 ⇒ δ = 1/100; Earth/Sun ratio 10,000
Sympy-exact symbolic witness of the transit-depth scaling δ = (R_p / R_s)² across representative ratios. Step 1 — hot-Jupiter-like R_p/R_s…
Doppler Δλ/λ = v/c: v = 100 m/s ⇒ shift 50/149896229; 51 Peg b 28/149896229
Sympy-exact symbolic witness of the non-relativistic Doppler relation Δλ/λ = v/c for stellar radial-velocity signals. Step 1 — canonical…
T_eq for (T_s, R_s/a, A)=(6000, 1/400, 0): 150√2 ≈ 212 K
Sympy-exact symbolic evaluation of the planetary equilibrium temperature T_eq = T_s · √(R_s / (2a)) · (1 − A)^{1/4} at a convenient…