PX282 - I2 - kepler's laws

first law

kepler's first law

a planet orbits the sun in an ellipse, with the sun at a focus of the ellipse

image: Carrol & Ostile, An Introduction to Modern Astrophysics, 2007

mathematically, an ellipse is described by:

r^{'} + r = constant = 2 a

where,
$r^{'}$ and $r$ are the distances from the foci to a point on the perimeter,
$a$ is the semimajor axis, which is the widest radius
also, $e$ , the ellpticity/eccentricity

considering a point, $Q$ , on the perimeter:

r^{2} = b^{2} + a^{2} e^{2} ⟹ b^{2} = a^{2} (1 - e^{2})

conic sections

eccentricity	shape
$e = 0$	circle
$0 < e < 1$	ellipse
$e = 1$	parabola
$e > 1$	hyperbola

all these shapes are allowed orbits

second law

kepler's second law

a line connecting a planet to the sun sweeps out equal areas in equal time intervals

image: Carrol & Ostile, An Introduction to Modern Astrophysics, 2007

third law

kepler's third law

the orbital period $(P)$ squared is proportional to the semimajor axis $(a)$ cubed

P^{2} = k a^{3}

therefore, $P$ gives the relative distance from the sun
eg: $P^{2} [yr] = a^{3} [AU]$
by measuring the parallax of transit of venus, the value of an astronomical unit was determined to $1 %$ of the modern value (1769,1771)
modern value is determined from radar measurements of venus