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…
Kepler third law T^2 = (4 pi^2 / GM) a^3; circular-orbit dynamic equilibrium
Kepler third law / circular-orbit framework (Kepler 1619 Harmonices Mundi V; Newton 1687 Principia Mathematica I.III.IV-VII). Setup: for a…
Photometric transit depth delta = (R_p / R_star)^2 from projected disk-area ratio
Transit photometry framework (Henry et al 2000 ApJ 529, L41; Charbonneau et al 2000 ApJ 529, L45; Mandel-Agol 2002 ApJ 580, L171). Setup:…
Hill sphere r_H = a (m_p / (3 M_star))^(1/3); 3-body tidal-self-gravity balance
Hill sphere of gravitational influence framework (Hill 1878 Researches in the Lunar Theory; Henon 1965 Annales d'Astrophysique 28, 992).…
Theorem: T^2 GM_star / a^3 - 4 pi^2 = 0 (Kepler third law residual)
Theorem (Kepler-third-law residual canonical): T^2 = 4 pi^2 a^3 / (G M_star) rearranges to T^2 G M_star / a^3 = 4 pi^2 exactly, so the…
Theorem: delta R_star^2 - R_p^2 = 0 (transit-depth disk-area-ratio identity)
Theorem (transit-depth-ratio canonical): delta = (R_p / R_star)^2 rearranges to delta R_star^2 = R_p^2, so the residual delta R_star^2 -…
Theorem: r_H^3 * 3 M_star - a^3 m_p = 0 (Hill-sphere cube-root identity)
Theorem (Hill-sphere-cube canonical): r_H = a (m_p / (3 M_star))^(1/3) cubed gives r_H^3 = a^3 m_p / (3 M_star), so r_H^3 * 3 M_star - a^3…
Mayor-Queloz 1995 51 Pegasi b discovery
Mayor-Queloz 1995 ELODIE radial-velocity detection of 51 Pegasi b: first exoplanet around Sun-like star; hot-Jupiter at 0.05 AU with…
Mandel-Agol 2002 transit photometry formula
Mandel-Agol 2002: analytical light-curve formulas for planet-transit including limb-darkening (linear, quadratic, nonlinear); standard…
First exoplanet (Mayor-Queloz 1995)
Mayor-Queloz 1995 (Nobel 2019) 51 Peg b first hot-Jupiter; modern 5500+ confirmed exoplanets via radial-velocity + transit.
Transit (Charbonneau 2000)
Charbonneau 2000 first transit detection HD 209458b; modern Kepler-2009 + TESS-2018 + JWST-2022 transit + transmission spectroscopy.
Microlensing (Mao-Paczynski 1991)
Mao-Paczynski 1991 + 1992 binary-lens; modern OGLE + KMTNet ~200 microlens-detected planets; bound + free-floating.
Direct imaging (Marois 2008)
Marois 2008 HR 8799 first direct imaging system 4 planets; modern GPI + SPHERE + JWST/MIRI direct-spectroscopy.
51 Peg b (Mayor-Queloz 1995)
M Mayor-D Queloz 1995 (Nobel 2019) 51 Pegasi b; first exoplanet-around-Sun-like-star; modern HARPS + ESPRESSO < 1 m/s.
Kepler mission (2009)
Borucki 2009 + Howell 2014 K2 Kepler-2,773 confirmed exoplanets; modern TESS 2018 + 5,500 confirmed by 2024 + JWST follow-up.
HD209458 (Charbonneau 2000)
D Charbonneau 2000 first transit + 2002 HD209458b atmospheric-Na; modern HJ-population + tidal-dissipation + obliquity studies.
Habitable zone (Kasting 1993)
J Kasting 1993 HZ; modern Kopparapu 2013 Earth-similarity + 3D climate-runaway-CO2-greenhouse + tidal-locking constraints.
Transmission spectra (Tinetti 2007)
G Tinetti 2007 first molecular-detection HD189733b H2O; modern JWST 2022 NIRSpec exoplanet-atmosphere routine spectra.
HR 8799 (Marois 2008)
C Marois 2008 HR 8799 4-planet direct-imaging; modern GPI / SPHERE / JWST + 2024 reflected-light Earth-imaging Roman-Habitable-Worlds.