# 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