w2 NewtonsVersionofKeplersLaw

Newton's Law of Gravity and Kepler's Laws

• Newton's law of gravity can be used to derive Kepler's laws of planetary motion.

• Key concepts discussed include:

  • Conservation of angular momentum.

  • Orbital speeds vary, with planets traveling faster at their closest point in orbit (periapsis) and slower at their farthest point (apoapsis).

Kepler's Laws of Planetary Motion

• Kepler's law relating the period of orbit and the separation of orbit can be expressed as:

  • ( P^2 \propto r^3 ) (P is in years, r in astronomical units).

• Important point: This relationship holds when the mass is similar to the mass of the Sun.

Correct Equation for Kepler's Law

• The more accurate version of Kepler's law is:

  • ( P^2 = \frac{4 \pi^2}{G(m_1+m_2)} a^3 )

    • Where:

      • ( P ) = orbital period in years.

      • ( a ) = separation in astronomical units.

      • ( m_1, m_2 ) = masses of the two objects.

• The equation simplifies under certain conditions, such as when one mass is negligible (e.g., when one mass is equal to the mass of the Sun).

Role of Constants

• ( 4 \pi^2 ) relates to circular motion and involves Newton's gravitational constant ( G ).

• ( G ) is given by:

  • ( G = 6.67 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{s}^{-2} )

• Consistency in units is crucial to correctly applying the formula.

Unit Consistency in Calculations

• It is important to use consistent units throughout calculations:

  • Example: If using kilograms for mass, distance should be in meters to obtain period in seconds.

  • If using grams for mass, ( G ) must be adjusted accordingly for appropriate consistency.

• Various unit conversions may be necessary in problems involving Kepler's law and gravitational calculations.

• Practice and examples will be provided in subsequent videos.