First Law — Law of Orbits

Planets move in elliptical orbits with the Sun at one focus.

Key points:

• Circle is a special case (eccentricity = 0)
• Sun at one focus; other focus is empty
• Semi-major axis a determines the size; eccentricity e determines the shape
• For Earth: eccentricity ≈ 0.017 (nearly circular)

Positions:

• Perihelion: closest point to Sun (planet moves fastest)
• Aphelion: farthest point from Sun (planet moves slowest)
• Earth is at perihelion around January 3, aphelion around July 4

Second Law — Law of Areas

The radius vector from the Sun to a planet sweeps equal areas in equal time intervals.

Mathematical form: dA/dt = L/(2m) = constant

Physical basis:

• Gravitational force is central (directed along radius vector)
• Central force → zero torque → angular momentum L is conserved
• Conserved L → constant dA/dt
• This is a consequence of angular momentum conservation (Noether's theorem)

Practical implication:

• At perihelion (minimum r): maximum speed
• At aphelion (maximum r): minimum speed
• Product r × v_perpendicular = constant at all orbital points

Third Law — Law of Periods

The square of the orbital period is proportional to the cube of the semi-major axis:

$T^{2}$ = (4π^{2}/GM) × $r^{3}$

So: $T^{2}$ ∝ $r^{3}$

Key applications:

• Ratio form: $T_{1}^{2}$/$T_{2}^{2}$ = r_{1}^{3}/r_{2}^{3}
• Finding period: T = T_Earth × (r/1 AU)^(3/2) years
• If r quadruples: T increases by 4^(3/2) = 8 times

NEET trap: It is $T^{2}$/$r^{3}$ = constant, NOT $T^{2}$/$r^{2}$ or T/$r^{2}$.