A good basic work calculation in two dimensions.

Good question. Under what conditions will an applied force do zero work? It is a geometric question, really.

This is important later when we consider angular momentum in an astronomical setting, where the axis of rotation is not necessarily the center of the orbit. In fact, an astronomical object like 'Oumuamua might not even be on a bound orbit, but an unbound orbit -- it enters the solar system, interacts gravitationally with the Sun and then exits the solar system.

Basic

Good.

Similar to the Bumblebee tilt test.

Good one.

See previous note.

We already knew this.

Same as the relationship between the propulsion force Fp from the road surface on the chopper and the friction force f, in the homework

Huh?

Yup, previous annotation is verified.

This is why I always say that intro physics is SO filled with right triangles.

Idealized force structure

Discussed this in lecture.

Rotational KE is simple to handle, but we will tackle it after Exam 3.

This is why scientists work most of quantum mechanics and relativity in terms of energies, not F = ma forces.

Work and change in kinetic energy, \(W=\Delta \left( KE \right)\)

Stopping distance problem, a classic exercise

True. And this is why physicists spend so much time thinking about spring/mass systems.<br> That is, what is simple to learn about spring/mass systems becomes widely applicable in other oscillatory systems like a beating heart, a particle inside an atom's nucleus, a light wave, a tsunami etc.

Atoms and guitars ← ← you will savvy this analogy in PHY2054C!!

Biggest lie in physics. "It can be shown that..." is usually the place where the author is thinking about calculus but does not want to actually show it.

ridiculous

Good example.

Another good example.

Extensive discussion about this Figure 3 in Friday's lecture.

All of these moments of inertia are the result of counting up the quantity \(mr^2\) for each pixel of mass, \(I=\sum_{\text{pixels}} \left(mr^2\right)\) To do this task, you usually need calculus, although the hoop can be done with straight trig, no calc.

Notice that each moment of inertia is a fraction or multiple of total mass \(M\) multiplied by the square or sum of squares of the overall dimension \(R^2\) or \(a^2+b^2\) etc. That is because each object in the table has some symmetry which simplifies the calculus and therefore simplifies the formula.

We will be demonstrating this stuff in lecture, 03/22/19

The National Advisory Committee for Aeronautics (NACA), where Hidden Figures' Katherine Johnson rose, was originated during WW I, was an intensive research facility focused on airframe optimization -- aerodynamic shaping. One famous airframe became the P51 Mustang, a prime fighter against the German Luftwaffe.

Katherine Johnson at NASA Langley

Non-relativistic, v << c.

$$F_{net}=ma$$

$$F_{net}=m \frac{\Delta v}{\Delta t}$$

$$F_{net}=\frac{m \Delta v}{\Delta t}$$

and for objects with constant mass, $$m \Delta v = \Delta \left(mv\right)$$ so $$F_{net}=\frac{\Delta p}{\Delta t}$$

Two good study questions.

Basic calculation

Generic wave equation. For electromagnetic radiation, the wave equation is \(c=\lambda f\), where the speed of light \(c=3\times 10^8 \frac{m}{s}\).

This is why Halley's successful prediction was so important.

It is enough for us to predict "larger \(\omega\)" or "smaller \(\omega\)" etc., and not necessarily calculate L to the nearest \(0.001 \,kg\,m^2\). After Exam 3, we will do some full calculations, however.

terrestrial and celestial, which at the time of Newton, were not universally considered as a unified system with universal laws governing both realms. In fact, the verification of Newton's law of universal gravitation simplified our view of the physical universe: one book to describe them all, as Galileo foretold.

Bypass, although it is an interesting section to read if you are interested.

Let's try this homework assignment: Experimental assignment.

$$c = \lambda f$$

Experimental homework

Bypass My balloon analogy in Mini-Lecture C is better.

Bypass

Bypass

Or, more simply, read it off the diagram.

Yes, definitely handy, because if we make a big telescope we can see and catch spectra of really distant, faint galaxies! E.g., from Kirshner, PNAS, 2004,

Lemaître was correct!!!! It forced Einstein to say that his own model of all galaxies was the biggest scientific mistake of his life.

There is a legend about this, that this relation came to Hubble as he drove down to Pasadena after a night's observing. He pulled his car over on the shoulder and stopped to think. It is a twisty mountain road; I have driven it myself. At some time later, hours later as the legend goes, a traffic cop pulled alongside to check him out. All was well -- he was just thinking of what it all meant, that the entire universe was expanding. Edwin Hubble was probably very late for his breakfast, and we still ponder this meaning today.

They were especially looking at the near-ultraviolet H and K lines of calcium, $$\lambda_H=396.8\:nm\longrightarrow\text{toward red, longer wavelengths}$$

$$\lambda_K=393.4\:nm\longrightarrow\text{also toward red, longer wavelengths}$$

Here is an image of the sun in a filter that only transmits the Ca K line, very purply blue.

So a galaxy's K line will be less purply blue, maybe an aqua blue or even green... i.e., shifted toward the red end of the Roy G. Biv spectrum

Hubble figured it out.

BINGO! An unexpected result.

How it was discovered

Bypass this. We will concentrate on Cepheid variables and Type Ia supernovas

I cannot find the original image on NASA servers. However, this image from 2014 is helpful for visualizing a Type Ia supernova: Image: Katzman Automated Imaging Telescope/LOSS

"Type Ia supernovae have acquired global importance in recent years through their use as distance indicators..."

So we have to be very alert to catch the peak intensity, "maximum light," before it starts to dim out.

Nice!

Handy cepheid variable stars!

For stars and galaxies, apparent luminosity (what we see on Earth) depends on its distance and its intrinsic luminosity (as measured at the galaxy or star itself).

Pretty technical -- we will bypass this section, too.

Interesting but we will bypass this section.

Nifty. This is like seeing a mega-Chelyabinsk event crush into the black hole... seeing it from halfway across the galaxy.

S2

First one with a good orbital track was S2, on a 15.2 year orbit about the black hole Sgr A*.

Cf., Schödel R. et al., "A star in a 15.2-year orbit around the supermassive black hole at the centre of the Milky Way" Nature 419, 694–696 (2002),

There are only a few stars within a parsec of our solar system.

Very hot. How was it heated to this temperature?

SNR

Definition of supermassive black hole. Some galaxies' central black hole is even larger than ours.

It works out to about \(R=46 \,LS\), which would be well inside Mercury's orbit, \(0.39\, AU=193\, LS\)

Gravitational lensing

Like the absorption features of the Sun, various colors picked out by the upper layers of the chromosphere and very hot atmosphere of the sun.

TWENTY TIMES!!!! Holy Toledo! Visible matter is like a drip from a Slurpee!

Astronomers were reluctant to accept this, but it is now undeniable.

There is a ton of fancy calculus behind the blue part of the graph. So the theory literally falls short of observations (red)

...the sign of "dark matter"

This would be like Neptune orbiting faster than Earth!!!

Yeah, what they SHOULD do.... ain't necessarily so.

E.g., a comet really moving fast at perihelion and slowing down at aphelion. E.g., Uranus and Neptune. Nearly the same mass, but orbital semimajor axes 19.2 AU for Uranus and 30 AU for Neptune; their orbital "years" are 84 years for Uranus but 164 years for Neptune.

Scientists must always be humble about their findings

$$M_{\text{Milky Way}}=1\times 10^{11}\,M_{\odot}$$

As in this diagram of a galactic year:

Physicists are always making assumptions like this, spherical galaxy, spherical planet, spherical this, spherical that. There is even a nerdish physics joke for which the punch line is, "Consider a spherical cow."

Physics humor. :\

Anyway, physicists do this to make things easier at the start, then they make more detailed, intricate models, as things progress

As previously stated, Kepler's Third Law is EVERYWHERE!!!!

: )

Galactic rotation curves fail, which is the sign of dark matter.

Big result, very tough to calculate.

As mentioned in the previous annotation

A big maybe. Scientists make large models with pixels representing stars of various masses and a gravitational interaction, then let the pixels evolve on the computer. This figure is a rough view of such simulations. A good simulation can be helpful, but they are very tough to make.

Wavelength \(\lambda = 21\, cm\) is a radio wavelength outside the FM band and the AM band.

E.g., Clearwater Lake Recreation Area, up in Ocala National Forest, about 31 miles northwest of UCF.

Helpful for comparisons.

Because of the spectral lines of heavier elements like calcium,

In other words, they are orbiting faster than they would if only the visible stars were present. The visible stars do not have enough mass to control these visible star clusters.

OH MY GOODNESS!!!!!!!

In the mountains above Los Angeles. Unfortunately, LA is too smoggy now and Mount Wilson is not such a hot observatory these days.

Just after the Revolutionary War, before the Constitution!

E.g.,

Musician and astronomer. NICE!

A good analogy.

We will tackle this kind of calculation in Module 3 when we study galaxies.

Cf., the famous Martian meteorite, ALH-84001.

But the Viking landers still gave us a lot of good data on Mars.

skip

Small but mighty, the good and faithful rovers that have sent us so much data about Mars.

Mysterious. Scientists definitely want to figure this out, if we are ever to colonize Mars

Compared to the Amazon River, this is shrimpy. The widest point of the Amazon is about 64 km.

This is a horrible sentence.

When a scientist deals with huge numbers, as in astronomy, it is sometimes "close enough" if they have the same power of 10 in scientific notation.

1. Mars polar cap \(= 1.00 \times 10^{15}\, \text{tons}\)
2. Greenland ice cap \(2.85 \times 10^{15}\, \text{tons}\)

So a better sentence would be this: \(\text{\color{blue}The mass of frozen water in the Mars polar cap }\)\(\text{\color{blue}is of the same order as the Greenland ice cap}.\)

One commonly hears egghead scientists saying that two quantities "are about the same order," and by that they mean the two quantities are righteous and equivalent.

Using \(\pi r^2\) for the area here is an underestimation, because \(\pi r^2\) is good for a flat circle, but the polar cap is not flat! It is curvature.

However, using \(\pi r^2\) is

1. easy
2. and close enough

We know this by looking for and measuring the infrared spectra of both \(H_2 O\) and \(CO_2\)

We would consider the dust clouds of Mars or of Earth (like in the Sahara) as different from the \(H_2 O\) clouds, which are microdroplets of liquid water held aloft by rising air currents.

Sublimation. Good comparison to what \(CO_2\) ice does here at the surface of Earth.

Standard 1.00 atmosphere of pressure on Earth = 1.01325 bars, or, as they say on the Weather Channel, 1013.25 millibars. This is the atmospheric pressure on a day of fair weather at sea level. A similar fair day in Denver, one mile altitude, would be less than 1013.25 millibars.

Mars atmospheric pressure is 0.007 bar or 7 millibars.

For comparison: The central pressure of a hurricane is considered extremely dangerous if it gets to 900 millibars. E.g., Hurricane Irma hit the Florida Keys in 2017 at Category 4, 929 millibars central pressure. Very violent.

Astronomers' preoccupation with \(H_2 O\)

Our pre-occupation with Mars, e.g., The War of the Worlds.

This is a huge mystery, the handedness of sugars and amino acids.

The other kind of molecule that astronomers are always alert for!!

like what a centrifuge does.

Isotopes tell us this age!

Cool!

Comets are an entire other subject!!

Excellent example is the Allende meteorite, which fell as a larger object that broke up, down in Mexico, on Feb. 8, 1969.

I.e., its quantum fingerprints!

Another redshift.

So \(\Delta \lambda = 0.3 nm\) which is in the numerator of the equation below.

I.e., \(\frac{\Delta \lambda}{\lambda}\)

Like the beautiful red \(H_{\alpha}\) line, for which the lab wavelength is \(\lambda=656.27\) nanometers.

i.e., in the lab on Earth

absorption OR emission lines can display redshift and blueshift

This is also the equation programmed into a policeman's radar gun, for measuring the speed \(v\) of speeders.

This means, as emitted by the source if it were in a stationary laboratory.

\(\Delta \lambda=\left(\lambda_{lab}-\lambda_{observed}\right)\)

So \(\frac{\Delta \lambda}{\lambda}\) is the percent change in wavelength.

Similar expressions exist for frequencies \(f_{lab}\) and \(f_{observer}\)

So a radio wave that has a smaller frequency than normal is redshifted. An xray wave with a higher frequency is blueshifted... all of this even though we do not perceive radio or xray as having color.

This is the main concept to focus on, for Doppler shifting. Redshift and blueshift are commonly used tools for studying galaxy clusters, galaxies, stars, planets and even atoms.

There are tons of volcanoes in the solar system. The largest volcano we know of is on Mars!!

Another example: the Himalayas. They formed when the subcontinent of India bashed northward into Eurasia.

Hmmmm...

Convection!!

Main source of forces and motion = convection. Huge blobs of molten lava convect from core to surface, like water boiling in a pot on the stove or a thunderstorm convecting water vapor and liquid water in the atmosphere.

or pinkish up in Georgia!!!!

Yup -- asteroids and comets are chunks of history going all the way back.

Good example, if you've been camping in Georgia, is the Pine Mountain formation, which includes a lot of quartzite.

I have a chunk of it on my desk in the Physical Sciences Building, something like this:

Consider, for example, the layers of sedimentary rocks in the Grand Canyon.

We will have a lot to say about Io.

Unlike some moons and planets which are geologically dead. Good example of dead geologically is our moon.

This means that water ice has a relatively high radar reflectivity. It does not dissipate the incoming radar beam, but bounces back a lot: strong return signal, "bright."

Astronomers are always looking for signs of water on planets, comets, asteroids, moons and on exoplanets.

Here is a freight train blowing its horn as it passes the rail fan's video camera. At about 0:39, the horn shifts from a high note (while approaching) to a lower note (while moving away from the camera).

This is a Doppler Shift for sound waves.

Electromagnetic waves -- radar, visible light, even xrays -- also experience Doppler shifting. It is how the sheriff's deputy nabs speeders with his radar gun, and it is how we can measure velocities toward or away from Earth.

Also using a radar rangefinder.

Using Kepler's third law and a radar rangefinder to actually get the orbit.

Also similar to Earth's core. However the metallic part of Earth's core is not nearly as big a fraction of Earth, compared to Mercury

Sand is a silicate mineral -- most white sands are principally silicon dioxide \(Si\:O_2\).

Earth also has iron/nickel core

Here is a set of diagrams that show the aphelion and perihelion in true proportion,

Study this section carefully: Basic info for using isotopes

Skim this section for basic information about the discovery of the structure of atoms. Prior to Rutherford, we though atoms were just blobs of something sprinkled with electrons.

Skim this section, because it relates to discrete spectra, one of our important tools in astronomy.

Conceptual definition of isotope. It is the number of protons that defines which element you have, but the nucleus can have any amount of neutrons it can hold onto.

IMPORTANT definition!!!!

This is the remarkable fact about atoms: they are mostly empty space! Makes a person think.

Which helps keep lava in its molten state!

Excellent graph and caption, visual definition of half life.

A lovely definition of half-life!

Like radioactive carbon-14 \(^{14}C\) spontaneously emitting an electron from its nucleus to become nitrogen-14 \(^{14} N\) which is stable. Most nitrogen, 99.632%, is nitrogen-14.

• Nucleus of carbon-14 = 6 protons, 8 neutrons
• Nucleus of nitrogen-14 = 7 protons, 7 neutrons. (The seventh proton used to be a neutron in carbon-14!)

Using the idea of "radioactive half-life" of an element.

Compared to Earth, where impact craters are rare, the Moon is loaded with craters. That is because Earth has weather, volcanoes, earthquakes and continental drift.

For Florida students, this term, snow, indicates a rare solid form of \(H_2 O\) which,at low temperatures, forms six-sided crystals that are very cold and fall from the upper atmosphere. Rare in Florida since the last Ice Age.

Important parts of the theory, though it is not yet a stone cold lock.

nicknamed Theia

Cf., mentions of oxygen isotopes in Lecture 8 Spring '18, (Edward D. Young et al. Science 2016;351:493-496) especially the relation of lunar materials to terrestrial rocks like the Mauna Loa lavas.

Cf., lecture 8, Spring '18

$$e=0.055$$

Cf., the NASA Space Science Data Center planetary fact sheet.

Yes, as the moon rocks from Apollo show. They have been analyzed for decades and compared to terrestrial rocks.

...where they are also heated up by the sun and become visible.

Important concept. They loiter out by their aphelion, but when they get in close to perihelion, they are really moving fast. So visible comets are only visible for weeks, if that.

Unlike the surface rocks most places on Earth. E.g., the Florida limestone bedrock which, in its oldest layers, is about 35 million years, less than 1% of the age of the solar system.

This is the part that always astounds me, that Kepler spotted this small amount of eccentricity!

"only" 11,000 mph!

about 107,000 mph!!

I usually use NASA's Planetary Fact Sheet. All sorts of neat data.

Similar to aphelion and perihelion but relative to Earth. You hear the guys at Mission Control in Houston talk about perigee and apogee. E.g., the ISS right now has perigee of 408 km altitude, apogee of 410 km altitude.

Important vocabulary terms: aphelion and perihelion.

Not labeled well, but compare to the Mini-Lecture A in YouTube.

We will discuss this after Exam 1.

There are objects that pass through the solar system on parabolic and hyperbolic orbits, such as the mysterious A/2017 U1 'Oumuamua (YouTube of its trajectory)

Focus on the subsection "Orbital Motion and Mass."

Bypass this calculation until end of the semester.

Here is the crucial strategy for astronomers!

• e.g., black holes
• e.g., binary star systems
• e.g., neutron stars
• e.g., galaxies orbiting each other like our galaxy and the Andromeda galaxy!

Here are the two masses:

• \(M_{sun}=1.9885\times 10^30 \,kg =1M_{\odot}\)
• \(M_{earth}=5.9723\times 10^24 \,kg = 0.0000030035 M_{\odot}\)

So in terms of \(M_{\odot}\), the terms \(M_1 + M_2=1.0000030035\) which is really close to 1.