Astronomy 217

 
 

Prof. Andrew W. Steiner

 
 
Sep. 8, 2021
 
 

TA James Ternullo

Last Time

  • Planetary Periods and Orbital Distances
  • Ellipses
  • Kepler's laws of planetary motion

Today

  • Newton's Laws
  • More on Vectors
  • Gravitational Force
  • Conservation
  • Angular Momentum
  • Centripetal Force
  • Inverse Square Laws

Kepler's First Law

  • The ellipse which describes the Earth-Sun distance (in polar coordinates) is $$ r = \frac{a (1-e^2)}{1+e \cos \theta} $$
  • The perihelion, at \( \theta=0 \) is the point closest to the Sun.
  • The aphelion, at \( \theta=\pi \) is the point furthest from the Sun.
  • For the Earth, \( e=0.017 \)

Newton's Laws

  • 1: An object in motion tends to stay in motion, and an object in rest, tends to stay at rest, unless that object is acted on by an external force
  • 2: $$ \vec{F} = \frac{d \vec{p}}{dt} $$
  • 3: "To every action there is an equal and opposite reaction; the mutual actions of two bodies upon each other are always equal"

Vector Review

  • A vector has a length and direction
  • Unit vectors have length 1, in "Cartesian" coordinates, they point along the axes $$ \vec{v} = v_x \hat{i} + v_y \hat{j} = 3.75 \hat{i} + 5.33 \hat{j} $$
  • Alternatively, polar coordinates describes vectors in terms of a length and an angle $$ \vec{v} = 6.52 \hat{r} + 54.9^{\circ} \hat{\theta} $$
  • Converting between the two: $$ r = \sqrt{x^2+y^2} $$ $$ \theta = \tan^{-1} (y/x) $$
  • Remember you can choose any origin you want!

Gravitational Force

  • Law of gravitation $$ \vec{F} = - \frac{G M m}{r^2} \hat{r} $$ where \( G = 6.674 \times 10^{-11}~\mathrm{N} ~\mathrm{m}^2~\mathrm{kg}^{-2} \)
  • Remember \( \mathrm{N} = \mathrm{kg}~\mathrm{m}~ \mathrm{s}^{-2} \)
  • Very weak force, except for very large masses

Inverse Square Laws

  • The gravitational force decreases as the square of the distance, \( F \propto 1/r^2 \)
  • Coulomb force (electrostatics) also works the same way
  • Because the surface area of a sphere is \( 4 \pi r^2 \), photons emitted from a star work the same way: $$ \mathrm{flux} = \frac{\mathrm{luminosity}}{\mathrm{area}} $$   $$ F(r_1) = \frac{L}{4 \pi r_1^2} $$   $$ F(r_2) = \frac{L}{4 \pi r_2^2} $$

Central Force

  • Gravity is an example of a central force.
  • More generally, a central force is a force for which the direction of the force between 2 objects follows the line that connects them and the magnitude of the force is a function of only the distance between them. Central forces are conservative.

The centripetal force which keeps Earth in orbit

Conserved Quantities

  • Key to understanding classical mechanics is the notion of conserved quantities or conservation laws.
  • Mass, sometimes viewed as conserved, but only approximately
  • Momentum: The total momentum, p = mv, of the system is conserved.
  • Energy: The total energy (in all forms) of the system is conserved.
  • Angular Momentum: The total angular momentum is conserved $$ \vec{L} = \vec{r} \times \vec{p} $$

Gravity and the Earth's Mass

  • Acceleration at the Earth's surface: \( g \approx 9.81~\mathrm{m}/\mathrm{s}^2 \)
  • This means we can obtain the mass of the Earth given the radius: $$ F = \frac{G M m}{R^2} = m g $$ thus $$ M = \frac{g R^2}{G} \approx 5.97 \times 10^{24}~\mathrm{kg} $$ with \( R_{\mathrm{Earth}} = 6.38 \times 10^{6}~\mathrm{m} \)
  • The average density is then just mass divided by volume $$ \bar{\rho}_{\mathrm{Earth}} = \frac{5.97 \times 10^{24}~\mathrm{kg}} {4 \pi (6.38 \times 10^{6}~\mathrm{m})^3} \approx 5.5 \times 10^{3}~\mathrm{kg}~\mathrm{m}^{-3} $$

Centripetal Force

  • Centripetal forces (circular motion) can be related to the velocity and the $$ F_{\mathrm{cent}} = \frac{m v^2}{r} $$ or in the Earth's case $$ F_{\mathrm{cent}} = \frac{M_{\mathrm{Earth}} v^2} {R_{\mathrm{Earth}}} $$ where $$ v = (2 \pi~\mathrm{AU})/(1~\mathrm{year}) $$

The centripetal force which keeps Earth in orbit

Mass of a Planet from Moon's Motion

  • Use Newton's laws plus circular motion, where \( m \) is the moon's mass, \( M \) is the planet's mass, and \( r \) is the distance between the planet and the moon $$ F = \frac{m v^2}{r} = \frac{G M m}{r^2} $$ where $$ v = \frac{2 \pi r}{t_{\mathrm{orbit}}} $$ and \( t_{\mathrm{orbit}} \) is the time it takes for the moon to orbit the planet

Vector Products

  • There are two vector products, a dot product which gives a scalar (number) from two vectors $$ \vec{v} \cdot \vec{w} = v w \cos \theta_{vw} $$ where \( v \equiv |\vec{v}| \), etc.
  • If two vectors are at right angles, their dot product is zero
  • And a cross product $$ \vec{a} \times \vec{b} = \vec{c} $$ We will often only have need of the magnitude: $$ c = | \vec{a} \times \vec{b} | = a b \sin \theta_{ab} $$ The direction of \( \vec{c} \) is determined by the "right-hand rule"

Angular Momentum

  • For a mass $m$ undergoing circular motion around some origin, $$ \vec{L} = \vec{r} \times \vec{p} $$ where \( \vec{r} \) is the vector from the origin of the circular motion to the mass $m$
  • Angular momentum conservation, energy conservation, and the \( 1/r^2 \) nature of gravity dictate the shape of astrophysical objects