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Extrasolar planets and Kepler’s Third Law 2 4 T2 a3 G M m AF. May 2015 Nicolaus Copernicus 1473-1543 Tycho Brahe 1546-1601 Johannes Kepler 1571-1630 Nose lost in 1566 following a sword duel with third cousin Manderup Parsberg over the legitimacy of a mathematical formula! Isaac Newton 16421727 r a 1 2 1 cos Polar equation of ellipse 2 b Eccentricity of 1 2 ellipse a 2 4 P2 a 3 Orbital period P G(M M ) Johannes Kepler 1571-1630 dA 1 2 G(M M ) 1 2 a dt Equal areas swept out in equal times This is a constant Radii of planets not to scale! Kepler’s Third Law Pa Orbital period P /years Johannes Kepler 1571-1630 P a Yr AU Mercury, Venus, Earth, Mars Saturn Jupiter Neptune 3 2 3 2 Uranus 2 4 P2 G M M a3 Semi-major axis of orbit a / AU Isaac Newton (1642-1727) developed a mathematical model of Gravity which predicted the elliptical orbits proposed by Kepler Planet and Solar masses Force of gravity r GMM F r2 G 6.67 1011 m3 kg -1s-2 a 1 2 1 cos Semi-major axis 2a M Polar equation of ellipse b 2 Eccentricity of 1 2 ellipse a 2 4 P2 a3 G(M M ) Semiminor axis 2b Orbital period P M r F GMM r2 Kepler’s Third Law T a f ( M , m) k days AU M 1.99 1030 kg G 6.67 1011 Nm 2 kg -2 k AU 1.49597871 10 m 11 24 3600s 1day 2 AU 3600 24 GM f ( M , m) 4 T a3 G M m 2 2 T 2 a 3600 24 AU days AU G M m T 2 AU days 3600 24 GM m planet mass 3 2 a M m AU M M 1 M m M M M star mass 3 2 3 2 365 3 2 3 2 T 2 a AU days 3600 24 G M m AU 3 2 3 2 T orbital period 3 2 a orbital 'radius' semi-major axis of elliptical orbit T a f ( M , m) k days AU k 2 AU 3 2 3600 24 GM f ( M , m) 3 2 365 M m M M Vertical intercept is very close to ln(365) = 5.90 Gradient is 1.49, not far off the Kepler #3 prediction of 3/2 T a 3 ln f ( M , m ) ln k ln 2 days AU k 2 AU 3 2 3600 24 GM f ( M , m) 365 M m M M f ( M , m) M m M M m jupiter 1.898 1027 kg m jupiter 1 M 1047.9 M star mass m planet mass f ( M , m) M star mass m planet mass M m M M m jupiter 1.898 1027 kg m jupiter 1 M 1047.9 M m M M M star mass f ( M , m) m planet mass m jupiter 1.898 1027 kg m jupiter 1 m m jupiter 317.85 m 0.0031 m jupiter 1 M 1047.9 M m M M M star mass f ( M , m) m planet mass m jupiter 1.898 1027 kg m jupiter 1 m m jupiter 317.85 m 0.0031 m jupiter 1 M 1047.9 Earth-like candidates?