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:
  $R = \frac{\sum m<em>i r</em>i}{\sum m_i}$

  • $r_i$ 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 $M<em>1$ and $M</em>2$ and distances $R<em>1$ and $R</em>2$ from the center of mass:
  $\frac{M<em>1}{M</em>2} = \frac{R<em>2}{R</em>1}$
  $M<em>1 R</em>1 = M<em>2 R</em>2$

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.

• $V_2 = \sqrt{\frac{2GM}{R}}$

• $V<em>2 = \sqrt{2} \cdot V</em>1$

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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