Laws of Kepler and Newton

Laws of Kepler and Newton

Introduction

  • Review of lectures by Pan Asen.

  • Focus on details needed for the International Olympiad on Astronomy (IOAA).

Third Law of Kepler and Newton's Law of Gravitation

  • Newton's Law of Gravitation (scalar form): F = G \frac{Mm}{R^2}

    • G is the gravitational constant.

    • M and m are the masses of the two bodies.

    • R is the distance between the centers of the two bodies.

Center of Mass

  • Formula for the radius vector of the center of mass:
    R = \frac{\sum mi ri}{\sum m_i}

    • $r_i$ is the radius vector of the i-th body.

    • $m_i$ is the mass of the i-th body.

  • If the origin is at the center of mass, then the radius vector of the center of mass is zero.

  • For two bodies with masses M1 and M2 and distances R1 and R2 from the center of mass:
    \frac{M1}{M2} = \frac{R2}{R1}
    M1 R1 = M2 R2

Derivation of Kepler's Third Law

  • Equating Newton's Law of Gravitation with centripetal force:
    G \frac{Mm}{R^2} = m a

  • Centripetal acceleration:

    a = \frac{V^2}{r} = 4\pi^2 \frac{R}{T^2}

    • r is the distance from the body to the center of mass.

    • R is the distance between the two masses.

  • Relating distances to the center of mass:
    \frac{M}{m} = \frac{R_1}{R}

  • R + R_1 = r

  • R_1 = \frac{mR}{M}

  • r = R + \frac{mR}{M} = R(1 + \frac{m}{M})

  • Substituting R into the force equation:
    G \frac{Mm}{R^2} = m \frac{4\pi^2 R}{T^2}

  • G \frac{Mm}{R^2} = m 4\pi^2 \frac{1}{T^2} \frac{R}{(1+\frac{m}{M})^3}

  • T^2 = \frac{4\pi^2}{GM}(M+m)r^3

  • T^2 = \frac{4\pi^2}{G(M+m)}r^3

Cosmic Velocities

First Cosmic Velocity

  • The velocity required for a body to orbit a planet in a circular orbit.

  • Derived by equating gravitational force and centripetal force.

  • V_1 = \sqrt{\frac{GM}{R}}

Second Cosmic Velocity

  • The velocity required for a body to escape the gravitational influence of a planet.

  • Derived by setting the total energy of the body to zero.

  • V_2 = \sqrt{\frac{2GM}{R}}

  • V2 = \sqrt{2} \cdot V1

Conservation Laws

  • Applicable in a closed system without non-conservative forces.

Conservation of Energy

  • In a closed system, the total energy is conserved.

  • Total energy = Potential energy + Kinetic energy

  • Potential energy:
    U = -\frac{GMm}{R}

  • Kinetic energy:
    K = \frac{1}{2} mv^2

  • Total energy at infinity is zero (assuming zero initial velocity).

  • The formula U = mgh is only valid when the gravitational acceleration is constant.

Conservation of Angular Momentum

  • The product of the moment of inertia and angular velocity is constant.

  • L = r \times p = r \times mv = mrVsin(\alpha) = constant

    • $\alpha$ is the angle between the radius vector and the velocity vector.

  • If the force is in the radial direction, the angular momentum is conserved.

Kepler's Second Law

  • Equal areas are swept out in equal times.

  • dS = \frac{1}{2}r(V \Delta t) sin(\alpha)

  • \frac{dS}{dt} = \frac{L}{2m} = constant

Radiation and Albedo

Luminosity

  • The total energy radiated by an object per unit time.

  • Energy flux at a distance R:
    F = \frac{L}{4\pi R^2}

  • Solar constant: The power of solar radiation per unit area at the Earth's orbit.

Energy Absorption

  • Energy absorbed by a body:
    E = \frac{L \pi r^2}{4 \pi R^2}

  • For objects close to a star, integration over the surface may be needed.

Albedo

  • The fraction of incident radiation that is reflected by a surface.

  • Radiated energy:
    E_r = \frac{A L \pi r^2}{4 \pi R^2}

  • Emitted radiation at distance R':
    F = \frac{A L r^2}{8 \pi^2 R^2 R'^2}

Vis-Viva Equation

  • Allows calculation of the velocity at any point in an orbit given the distance to that point.

  • V^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right)

    • V is the speed of the body.

    • G is the gravitational constant.

    • M is the mass of the central body.

    • r is the distance between the bodies.

    • a is the semi-major axis of the orbit.

  • Derived from conservation of energy.

Virial Theorem

  • Relates the average kinetic energy of particles in a system to the forces acting within the system.

  • 2 \langle T \rangle = n \langle U \rangle

    • $\langle T \rangle$ is the average kinetic energy.

    • $\langle U \rangle$ is the average potential energy.

    • n is related to the force (related to the inverse square law of gravity).

  • For gravitational forces (U = -\frac{GMm}{R}), n = -1. Then:
    2 \langle T \rangle = -\langle U \rangle

  • For a gravitationally bound system:

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