Inverse Square Law Force

For gravitational force:

$$ F = -\frac{k}{r^2} $$

Orbit Equation

$$ \frac{d^2 u}{d\theta^2} + u = \frac{k}{mL^2} $$

Solution

$$ u = A \cos\theta + \frac{k}{mL^2} $$

Orbit Shape

$$ r = \frac{1}{A \cos\theta + \frac{k}{mL^2}} $$

This represents a conic section:

• Ellipse
• Parabola
• Hyperbola