Kepler’s First Law (Law of Orbits)

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Kepler’s First Law states:

“The orbit of a planet is an ellipse with the Sun at one focus.”

This law describes the geometric shape of planetary orbits and defines how celestial bodies move under gravitational attraction.

Mathematical Form of the Ellipse

An ellipse in polar coordinates, with the central body at one focus, is described by:

r(\theta) = \frac{p}{1 + e \cos(\theta)}

where:

  • r is the distance from the central body (focus) to the orbiting body,
  • e is the eccentricity (0 \leq e < 1 for ellipses),
  • p is the semi-latus rectum, related to the orbit’s size and shape.

Geometric Properties of the Ellipse

  • Semi-major axis a: The longest radius, half the longest dimension across the ellipse.
  • Semi-minor axis b: The shortest radius, perpendicular to the semi-major axis.
  • Relationship between a, b, and e:

b = a \sqrt{1 - e^2}

Foci:

The ellipse has two foci, separated by distance 2ae. The central body (e.g., the Sun) lies at one of these foci.

Physical Meaning

  • The planet does not orbit the center of the ellipse, but one of its foci.
  • Distance from the central body varies throughout the orbit:
    • Periapsis (closest approach): r_{\text{peri}} = a(1 - e)
    • Apoapsis (farthest distance): r_{\text{apo}} = a(1 + e)
  • This varying distance leads to changes in orbital speed, explained by Kepler’s Second Law.

Special Case: Circular Orbit

  • When e = 0:

r = p = a

The orbit is a circle, and the central body is at the center.

Examples in the Solar System

  • Earth’s orbit has e \approx 0.0167, nearly circular.
  • Comet orbits often have high eccentricities, producing long, thin ellipses.

Significance

Kepler’s First Law was revolutionary because it replaced the ancient idea of perfect circular motion with elliptical motion. It provided the geometric foundation for understanding planetary paths and set the stage for Newton’s gravitational theory.

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