# Tag Archives: physics

## Explore the Science behind the Ghostbuster’s Proton Pack

The new Ghostbusters (2016) features a major upgrade to an iconic device: The Proton Pack. In the original movie, the Proton Pack was a portable particle accelerator that emitted a stream of positively-charged protons to ensnare negatively charged spiritual entities. This device was based on one of the earliest particle accelerators: the cyclotron.

Jefferson Particle Physicist and Ghostbusters Science Consultant James Maxwell

## The Cyclotron

Basic model of the Proton Pack showing as cyclotron.

The cyclotron was one of the earliest types of particle accelerators ever developed. Invented by Ernest O. Lawrence in 1932, it works by accelerating charged particles along a spiral path. Inside the cyclotron, a charged particle is injected into the middle of the chamber where it is accelerated between two D-shaped electrodes or “dees.” In the case of the Ghostbusters’ Proton Pack, that charged particle is a positron or positively charged electron. The magnetic field passing through the dees bends the particle’s path, making it travel in a circle, while the electric field between the dees accelerates the particle, giving it a “kick” to make it go faster. When the positron beam has enough energy, it strikes a metal target to release a beam of protons.

## The Synchrotron

The upgraded Ghostbusters’ Proton Pack features a more advanced particle accelerator: the Proton Pack

The upgraded Proton Pack is based on the synchrotron. Like the synchrotrons that make up many of the modern-day particle accelerators, this allows for higher energy particle streams. The idea behind the upgrade was due to the movie’s science consultant, and Thomas Jefferson National Accelerator Facility particle physicist, James Maxwell.

## Crossing the Proton Pack’s Beams

Fighting ghosts on the Ghostbusters

The movie’s writers, Paul Feig and Katie Dippold have put considerable effort into getting the science right. It is definitely worth a look at the science behind the Proton Pack.

Article: The Science of The Ghostbuster’s Proton Pack.

## Flying West or East. Why is one faster?

In his Minute Physics video, Henry Reich looks at the somewhat paradoxical question of why is it faster to fly west than east. Given that the Earth rotates from west to east, we expect that a plane flying west would get to its destination faster as their destination is moving towards them. Think the person or place you are running to is also moving towards you–the relative speed increases. This does not happen. Instead, planes take longer traveling west along the same route.

The speeds add when the plane travels to the East (left) but they subtract when the plane travels West (right).

The answer to this question lies in the Earth’s rotation. Reich explains that as the Earth rotates, everything rotates along with it at a speed of 1180 km/h along the Equator. This includes the objects on the ground, as well as the air above. A plane generates lift by moving relative to the air around it, and most planes can travel anywhere between 500 to 900 km/h. This means that an airplane traveling East, along the Earth’s rotation, will travel faster because their speeds add. In this case, the plane is traveling 1180 km/h + 500 km/h = 1680 km/h relative to the ground. If the plane travels in the opposite direction, the speed relative to the ground is less because we subtract the speeds — 1180 km/h – 500 km/h = 680 km/h. We do this because we must take the directions that the Earth and the planes move. In Physics, we know this as vector addition.

Of course, that is the simple answer, and as Reich illustrates in his video, there is far more to the answer. The fact that the air moves, either pushing a plane along or pushing against it, has everything to do with the Earth’s rotation. The prevailing direction of these winds or the jet stream is due to the Coriolis Effect, the apparent deflection of the winds is due to the fact that the Earth rotates. This means that airplane travel times are influenced by the Earth’s rotation. The reason is not as straightforward as we would like.

## Metamaterials and the Science of Invisibility

Invisibility is just one of the many features of Kiera Cameron‘s (Rachel Nichols) City Protective Services (CPS) uniform. A cloaking device, such as the one used by CPS officers, deflects light around it rendering it invisible. Such a device would be based on metamaterials; an artificial material specifically engineered to have properties not be found in nature.

Prof. Sir John Pendry, the father of metamaterials.

The idea for such a material was first proposed by English physicist, Sir John Pendry in 2006 in a paper published in Science, “Controlling Electromagnetic Fields”. In this paper, Pendry describes how electromagnetic fields can be redirected around an object thereby rendering it invisible.

The way a material affects light is largely determined by its chemical composition. A metamaterial is different because its properties are derived from its physical structure; there are repeating microscopic patterns on the surface. Scientists can engineer this structure to bend light around an object thereby rendering it invisible.

HMS Argus using dazzle camouflage in 1918

Becoming invisible has always been an important part of military strategy since the 18th century and the emergence of the long-range rifle. Soldiers would camouflage themselves by either dressing in forest green or field grey. During World War I, troops started to experiment with “dazzle camouflage”.

The striped pattern of the dazzle camouflage is a poor choice to hide something as it draws attention to a person or an object. The pattern does have one advantage–it makes it difficult to estimate an enemy’s range. This concept was soon used on ships as it made it difficult for an observer to know exactly whether the stern or the bow was or whether the ship was moving towards or away from an observer’s position.

Dazzle camouflage continued to be used until World War II but its effectiveness was severely limited with the introduction of radar. The Germans sought to hide their craft with the use of radar-absorbing materials, the earliest of which was used on submarine periscopes and consisted of a layered coating of graphite particles and other semiconducting materials embedded in a rubber matrix. This material was effective at reducing reflections in the 20 cm radar band range used by the Allies.

The Horten Ho’s flying wing design made it the first stealth aircraft.

The Germans went further by incorporating carbon-impregnated plywood in the Horten Ho 229–the first ever pure flying wing powered by jet engines. It was believed that the carbon powder in the glue would shield the aircraft from detection by the British early warning ground-based radar, known as Chain Home.

Carbon is a cheap material with low conductivity. This makes it possible tailor conductivity from synthetic materials made from the element. Conductivity can be very poor (if made from insulated grains of carbon black or soot) or very high (if made from connected chains of graphite). When an electric field encounters a carbon-based absorber, it induces electrical currents in the material which are then dissipated as heat.

## Structure Dependent Properties

The hint that a material’s shape and structure at the nanoscale could affect its properties came in the 1990s. The British company, Marconi Materials Technology manufactured a carbon material capable of hiding battleships from radar but had no idea how it worked. They approached John Pendry to find the answer. Pendry discovered the electrical properties that allowed the material to absorb radiation didn’t come from the carbon itself but from the shape of its long thin fibers.

This was a significant discovery. Instead of changing a material’s chemistry to alter its behavior, scientists could alter the internal structure at very fine scales–smaller than a wavelength of light–to get the same effect.

All electromagnetic radiation has two components: a magnetic field and an electric field. As an electromagnetic wave strikes a material the atoms respond like a tiny magnet–its electrons move in a circle in response to the magnetic component and back and forth in response to the electrical component.

Each split-ring is designed to respond to the electromagnetic field in a certain way. When put together in an array with other split-rings, the periodic construction of many of these cells interacts with the electromagnetic wave as if these were homogeneous materials. This is similar to how light interacts with everyday materials, e.g. glass.

We can also create a magnetic field by looping a current around a circle. This magnetic field is more concentrated in the center of the loop than outside. Pendry hypothesized that by creating loops of a non-magnetic material, such as copper, he could create a similar magnetic response typically found in magnetic materials. Scientists would be able to tune how electrons move by tuning the size and shape of these loops. This controls how incoming radiation is bent when it encounters an object.

Pendry wondered how far this new insight could go. Would it be possible to change the magnetic properties of a material by simply changing its fine structure alone and not its chemistry? If so, then a theoretical non-magnetic metamaterial could mimic some of the properties of a magnetic substance like iron.

A split-ring resonator array constructed using copper split-ring resonatorss and wires mounted sheets of fiberglass circuit boards. The copper rings respond to the magnetic component while the mounted wires respond to the electroc field of an EM wave.

Pendry thought of taking this a step further. By cutting the loops, he created what is known as a magnetic resonator that acts like a switch. This switch would allow him to change a metamaterial’s magnetic properties on command. In so doing, by combining what he learned from Marconi’s radar absorbing material he figured a way to manipulate electromagnetic radiation. This makes invisibility possible for Continuum’s CPS officers.

## Bending Light around Objects

An example how an object appears invisible using mirrors.

Invisibility can be considered the supreme form of camouflage as it does not reveal anything about an object to an observer. We can accomplish this using a plane mirror and two parabolic mirrors to reflect light around an object. The object becomes invisible from two sides.

Metamaterials make an object invisible by bending light around it. (Image from Pendry et.. al. Science (2006): 1780-1782).

A metamaterials invisibility cloak will work in almost the same way as the parabolic mirrors by steering radiation around an object. The many tiny elements of a metamaterial–the fine scale structures–pick up rays or light from the far side of an observer and relay that ray around the material. When the ray arrives at the side facing the observer, it is re-emitted in the direction it would have taken as if the object was not there at all. Unlike the parabolic mirror trick, a metamaterial cloak will have to do so in all directions. To do this, we need a three-dimensional array of metamaterials

The fishnet metamaterial could one day become the invisibility cloak of the future.

One way of accomplishing this is to create a “fishnet”. In this case the metamaterial is made of alternating sheets of glass and silver containing rectangular holes. This design was developed in 2008 at the University of California Berkeley. As light travels through the fishnet, the alternating layers bend light in unusual ways. The research group at Berkeley hope that this array will eventually be able to guide visible light around an object.

## Other uses of Metamaterials

The use of metamaterials extend beyond manipulating the electromagnetic spectrum. It can also be used to create acoustic and tactile cloaks, preventing a user from being heard or felt. The acoustic cloak is made from perforated plastic sheets arranged in a pyramid. When it is placed over an object, sound waves act as if nothing is there, as if there was only a flat surface in their path.

The sound cloak is made of perforated plastic sheets in a periodic pattern.

The acoustic cloak alters a sound’s path in the same way the invisibility cloak does. Sound doesn’t penetrate into the pyramid but is rerouted in a way to create the impression that noting is there.

While it may one day be possible to completely hide future CPS officers in the visible spectrum and from being detected by sonar, what happens when they bump into something or, per chance, someone bumps into them? Metamaterials can also be designed to create a mechanical cloak.

Metamaterials prevent the object at the bottom from being felt.

Like the acoustic invisibility cloak, the tactile cloak is made from a periodic polymer array, the properties of which are determined by its special structure. AN object placed under a blanket or layers of foam would still be felt under the blanket. When an object is placed under this “cloak” it redirects stress in such a way that its shape can’t be felt when the cloak is touched.

## The CPS Invisibility Suit: Is it possible?

The fishnet design has the advantage that it can handle a wide range of wavelengths; previous designs could only cloak at a specific wavelength.There are some disadvantages, however. It only works on flat surfaces and not the sleek CPS uniform we see Kiera wearing.

There is another limitation to wearing an invisible CPS uniform: an officer won’t be able to see out for the same reason you can’t see in. While the suit is invisible in a certain range of wavelengths–the visible spectrum–it is visible in other areas of the spectrum.

We know that a CPS officer’s cybernetic visual implants allow them to see in other parts of the spectrum, in the infra-red and possibly ultraviolet range. In a previous article, we discuss this ability may be based on graphene contact lens, a hexagonal mesh of carbon atoms. A CPS officer can activate their visual implants to see in other parts of the spectrum when they turn invisible.

We can only imagine what tech is hidden in Kiera’s CPS uniform. Though she has revealed some of her suits abilities to us we can only imagine what more she can do.

## Godzilla 2014: How Can The MUTO Talk? Communication and Echolocation

MUTO meet to mate

In the 2014 Godzilla reboot, Dr. Joe Brody discovers that the M.U.T.O. (Massive Unidentified Terrestrial Organism) may be using a form of echolocation to communicate with another of its kind hidden in the Yucca Mountain Nuclear Waste Depository in Nevada. How exactly can a creature in Japan communicate with another all the way in Nevada? It uses ultrasound. To learn more, read my new MoviePilot article.

## The Science of Godzilla’s Roar Does he really purr like a cat

There is nothing as intimidating or awe-inspiring as the most famous Kaiju’s roar. Though fictional, this is one of the things that defines him other than his massive size. But there are problems were physics is concerned. Sound is generated by the larynx in one of two ways: either purring mode or flow-driven mode. Purring mode depends on how quickly the vocal folds or vocal cords contract and vibrate. Flow-driven mode depends on how passing air vibrates those folds.

The sound produced in flow-driven mode depends on the size of the larynx. The larger the larynx, and hence the vocal folds, the lower the sound. Godzilla is so large, his roar would be in the infrasonic range; below the range of human hearing. Purring mode suffers from no such limitation. Could the King of Monsters actually be purring? Read my MoviePilot article to learn more.

## The Discovery of the Higgs Boson: Week 2 Review Light Bulbs and Railroad Schedules

The second week of the FutureLearn course “The Discovery of the Higgs Boson” looks at physics of the 20th century–special relativity and quantum mechanics. These two branches of physics represented a fundamental shift in the way we view the world.

It may come as a surprise to some that these deep philosophical shifts have very unexpected origins. Our view of a quantized world came from the need to create a more efficient light bulb while the connection between space and time came from our need to run a more efficient railroad network and international time conventions.

## Need for a more efficient light-bulb

Hungarian physicist Philipp Lenard, discoverer of the photoelectric effect in 1902.

In 1902, Hungarian physicist, Philipp Lenard, winner of the 1905 Nobel Prize in Physics for cathode rays, observed that the energy of individual emitted electrons increased with light frequency–the photoelectric effect. This appeared to be at odds with Maxwell’s theory of electromagnetism which predicted that an electron’s kinetic energy should be proportional to light intensity. In 1905, Albert Einstein published a paper that explained the experimental data from the photoelectric effect. Based on Max Plank’s theory of black body radiation Einstein postulated that light energy was being carried in discreet quantized packets.

In 1894, German theoretical physicist Max Planck was commissioned by the German Bureau of Standards with the task of creating more efficient light-bulbs. To do so, Planck needed to find one that would emit as much visible light as possible with very little to no infra-red and untra-violet light. Planck knew from experiments at when an object is heated, it emits radiation in the form of black-body radiation. Planck turned his attention to this problem.

Black body curves for various temperatures and comparison with classical theory of Rayleigh-Jeans. As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths.

“Blackbody radiation” or “cavity radiation” is the characteristic radiation that a body emits when heated. This is seen in the form of a curve which peaks at a characteristic temperature where most of the radiation is emitted. Experiments showed that as the temperature changes, so too does the emitted radiation. When the wave picture of light was applied to this problem, it failed to predict the observed intensity for any given temperature.

Planck made several attempts to understand this problem. His first proposed solution in 1899 based on the entropy of an ideal oscillator, in what he called the “principle of elementary disorder”, failed to predict experimental observations. Planck revised his approach in 1900 using Boltzmann statistics to gain a more fundamental understanding of black-body radiation. This approach worked but Planck held an aversion towards statistical mechanics. He was also deeply suspicious of the philosophical and physical implications of its interpretation. His recourse was, as he later put it, “an act of despair… I was ready to sacrifice any of my previous convictions about physics.”

The central assumption behind his third attempt was the hypothesis, now known as the Planck postulate, that electromagnetic energy could only be emitted in quantized form. Planck didn’t think much of this method, regarding it as a mere trick. We know now that assumption is regarded as the birth of quantum mechanics. Try as he might, Planck struggled to grasp the meaning of energy quanta, going so far as to reject Einstein’s hypothesis and explanation of Lenard’s photoelectric effect. He was unwilling to completely discard Maxwell’s theory of electrodynamics.

Not everyone was convinced by Einstein’s hypothesis either, even after it was experimentally verified by Robert Millikan in 1914. Many physicists were reluctant to believe that electromagnetic radiation could be particulate in nature. Instead, it was believed that the observed energy quantization was the result of some constraint of matter and the way that it absorbs and emits radiation. It wasn’t until Compton’s experiments showed that light cannot be purely be explained as a wave that the idea of light quanta was accepted.

## Train Schedules and Time Zones

The first passenger carriage in Europe, 1830, George Stephenson´s steam locomotive, Liverpool and Manchester Railway

The mid to late-19th century saw considerable and rapid improvements in transportation, communication and technology. One of these inventions, the steam locomotive, not only changed the way goods and people traveled but also the way we view time. Products could be moved more cheaply and much faster.

Before the invention of clocks, people marked the time of the day with apparent solar time or by noting the sun’s position in the sky. Local time was different for each town and settlement. With the invention of well regulated mechanical clocks, cities used local mean solar time. As clocks differed between towns by an amount corresponding to the difference in geographic longitude–a variation of four minutes for every degree of longitude–communication between towns and rail transport became awkward. The time difference between Bristol and London, for example, a difference of 2°35′ longitude, is about 10 minutes while the difference between New York and Boston is about two degrees or 8 minutes.

Time keeping on American railroads was even more confusing. Each railroad had their own standard of time time, usually based on the local time of its headquarters or main terminus. Each railroad schedule was published using the company’s own time and stations had a clock for each railroad, each showing a different time.

Non-uniform time zones weren’t just confusing. It was dangerous. The incidents of accidents and near-misses became more frequent as more people started using trains for travel. What was needed was a means to know exactly where trains were at all times. The use of time zones solves this problem and with it came the need to synchronize clocks at a distance.

It’s not surprising that many of Einstein’s though experiments concerns trains. As a young patent clerk, many of the inventions he reviewed focused on using light signals to synchronize clocks. Einstein took it a step further and realized that clocks moving with respect to each other would not tick at the same rate.

Physics students are familiar with Einstien’s Gedankenexperiments and that the power of abstract thought can allow one to fully visualize the consequences of an experiment without having to actually perform said experiment. Far from being esoteric examples, Einstein’s thought experiments are firmly grounded in reality and shares its origins in something as simple as a train schedule.

## The Higgs Boson Course

In one of the course’s lectures, Peter Higgs says that when he teaches undergraduates special relativity, he ignores the way that Einstein did it and asks, “how do you realize the principle of relativity, which was what was formulated by Henri Poincare?” To do this, you have to abandon Newton’s assumption of absolute time. Peter Higgs is correct, the development of special relativity need not have had anything to do with the Michelson-Morley experiment. In Einstein’s case, it came about from the practical need to synchronize clocks.

The second week of the course builds on the previous week. Though the concepts are quite literally mind-blowing, the ideas and mathematics were conveyed in a way that makes it easy for students to grasp. The third week looks even more exciting as we combine both special relativity and quantum mechanics to make much deeper predictions about our world.

## The Discovery of the Higgs Boson: Week 1 Review Conservation Laws and Physical Revolutions

The Higgs boson has captured the world’s attention from the moment it was announced the Large Hardron Collider was being built to find it. The elementary particle’s discovery was announced by CERN on 4 July 2012 and is nothing short of monumental as it appears to confirm the existence of the Higgs field. This pervasive field is pivotal to understanding why some fundamental particles have mass. As interesting and exciting as this discovery may be, its consequences and implications remain out of reach for the general public. Future Learn, a privately company owned by Open University, along with the University of Edinburgh have started a seven week Massive Open Online Course (MOOC) “The Discovery of the Higgs Boson” to introduce the theoretical tools needed to appreciate this discovery.

The course starts with classical mechanics. While it may seem like a strange place to start, especially given the course’s goals, it is a good one. Rather than jumping straight into Quantum Mechanics, classical mechanics makes the mathematics is accessible to students, especially those who have completed A-Level Mathematics or done Calculus. The course doesn’t assume many of the fundamental conservation laws in physics are true but rather goes through the mathematically rigorous process of proving these concepts to students. This also allows students to see how the behavior of physical systems can be deduced. Though some of the physical principles will need to be revised as the course progresses, the mathematical tools students would have acquired remain the same. Student can thus build on what they have learned before.

## An Evolutionary Revolution

One of the more interesting questions asked was, “Why is the Higgs boson discovery important?” Dr. Victoria Martin, a reader in particle physics at the University of Edinburgh, answers that in one of the course’s modules. In her video, she says the Sun is a massive burning ball of hydrogen and helium and it is a mystery why it all hasn’t burned up by now. She says the answer comes from Peter Higgs’ theory which predicts the Higgs boson. It predicts why the nuclear process in the Sun is slow enough that the Sun is still around after 4.6 billion years and provides just the right amount of light and heat to sustain life on Earth.

Science didn’t always believe that the Sun or the Earth, for that matter, was old. In the mid-nineteenth century, both Charles Darwin and Alfred Russel Wallace were making the case for biological evolution by natural selection. This theory described a process of slow, gradual changes over time and indicated the Earth had to be very old, at least hundreds of millions of years. This was supported by geologist observations of erosion rates.

Photograph of William Thomson, Lord Kelvin.

This posed a problem for the most prominent theoretical physicist of the time, William Thomson, 1st Baron Kelvin, who saw evidence that disagreed with Darwin. This guiding light of the Industrial Revolution, whose work in thermodynamics contributed to the steam engine, was a devout Christian who believed in a much younger Earth and with good reason. In 1862, using the thermodynamics of heat conduction, Thomson initial calculations showed that it would take between 20-400 million years for a molten Earth to completely solidify and cool.

This large uncertainty of the Earth’s age were due to uncertainties about the melting temperature of rock. This did not deter Thomson who then set out to calculate the Sun’s age in 1868 using what he knew of the Sun’s energy output. Kelvin rightly assumed the Sun formed from a giant gas cloud and gravity eventually caused the cloud to collapse into a ball. As with any falling mass, the cloud molecules’ potential energy would be converted into kinetic energy. This raise in kinetic energy would turn into heat, raising temperatures to result in star formation in a process known today as the Kelvin-Helmholtz Contraction.

While we know today this is not the way stars generate all their energy, we know this is how the fusion process is started. Based on his assumption that the Sun built up all its heat as it was formed and radiating it away like a hot coal, Kelvin estimated the lifespan of the Sun to be about 30 million years.

The numbers posed a nagging contradiction–the Earth was older than the Sun. Thomson realized he needed to refine his calculations of the Earth’s age. In 1897 Thomson settled on an estimate that the Earth was somewhere between 20-40 million years old. This fit in nicely with his estimation of the Sun’s age.

Darwin, aged 45 in 1854, by then working towards publication of On the Origin of Species.

The age of the Earth was an important part to Darwin’s theory of evolution. As a geologist, he had conducted his own studies and concluded that the time needed to wash away the Weald, a valley in south-east England formed of the eroded remains of an anticline, would require 300 million years. Thomson believed that geologists were wrong to assume a steady rate of erosion. Floods and other natural disasters could accelerate this process. Thompson though the geologist’s thinking could use a dose of mathematical rigor. Though Darwin’s observations supported an old Earth, he was so bowled away by Thomson’s analysis that he removed any reference to time scales in later editions of his Origin of the Species.

## A Quantum World

Thomson’s opponents argued that his time scales were too short for life to develop. He ignored them. With hindsight, it is easy to see that the brilliant Thomson was wrong and Darwin was right. But should we judge Thomson harshly for not listening to his opponents?

We must be careful when we judge the past and not look through the lens of our own experiences or biases. Thomson lived to 1907, to a time when radioactivity had firmly been established. The observation then by geologists that the Earth could be heated from within by radioactive decay meant that the Earth could be a lot older than Thomson thought. In fact, it was widely believed that the discovery of radioactivity invalidated Thomson’s estimates of the age of the Earth.

This image shows the Sun as viewed by the Soft X-Ray Telescope (SXT) onboard the orbiting Yohkoh satellite

Despite the discovery of radioactivity, Thomson refused to acknowledge this. He had strong reason to believe that the Sun was younger than 20 million years. Even with an old Earth, without sunlight, there could be no explanation for the sediment record on the Earth’s surface. It wasn’t until the discovery of fusion in the 1930s that this paradox was resolved.

While history has proven Thomson wrong in this debate, we must remember one thing. Given what we knew at the time, Thomson’s calculations and conclusions were correct–his science was sound. We can not fault Thomson for this.

## The Higgs Boson Course

Data from the CMS experiment, one of the main Higgs-searching experiments at the Large Hadron Collider. Image: CERN

The Higgs boson is the latest addition to our understanding of what happens in our Sun. Just like Thomson demanded a certain mathematical rigor, so too does “The Discovery of the Higgs Boson” course. The course recommends that the week’s module should take about two hours. Though it has been some time since I last sat in a Physics class and while the concepts and proofs were not entirely new, it does take some time to view the lectures and complete the exercises. I think students will realistically have to dedicate more than two hours to complete the week’s exercises.

Overall, it is a strong course that demands a lot of its students. I look forward to the rest of the course.

## The Big Bang Theory of Trojan Asteroids

Season 06, Episode 03: “The Higgs Boson Observation”

In “The Higgs Boson Observation”, we learn that the subject of graduate student and Sheldon’s (Jim Parsons) new assistant, Alex Jensen (Margo Harshman), is on Trojan asteroids at the Earth-Sun L5 Lagrange point. This also happens to be Raj’s (Kunal Nayyar) area of expertise. But what exactly are Trojan asteroids and why are they of interest to astronomers and astrophysicists?

Trojans are minor planets or satellites that share an orbit with a planet but does not collide with it because it orbits around one of the two stable Lagrangian points (trojan points). The L4 and L5 lie approximately 60° ahead of and behind the larger body, respectively .

# Area of Research

These are the five Lagrangian points in a two-body system with one body far more massive than the other (e.g. the Sun and the Earth)

There are five points on or near Earth’s orbit known as Lagrange points where an asteroid will remain stationary with respect to Earth as it orbits around the Sun. These points mark the positions where the combined gravitational pull of two large bodies, such as the Earth and the Sun, provides the centripetal force needed to orbit with them.

The Lagrangian points are approximate solutions to what is known as the three-body problem, a famous problem that attempts to model the motion of three bodies as they gravitationally affect each other. In essence, this problem dates back to 1687 when Sir Isaac Newton first published his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), often referred to as simply the “Principia”. Several scientists and mathematicians have all attempted to find an analytical solution to the problem or a simple equation that would describe the motions of all three bodies.

The three-body problem is difficult to solve because the three bodies tug against each other in chaotic and unpredictable ways. Only by simulating the problem on a computer can we see the objects paths. In comparison, the quantum three-body problem is relatively easy to solve and understand. However, approximate solutions can be useful. If the third mass is taken to be small enough that it does not affect the other two masses some interesting solutions crop up.

# Celestial Mechanics

To better understand how celestial bodies move in our solar system, we turn to Newton’s law of universal gravitation. This law states that every mass in the Universe attracts every other mass with a force that is proportional to the product of their masses. If we had two masses, $$M_{1}$$ and $$M_{2}$$, then the force between them would be proportional to the product of the two masses.
$F \propto M_{1}M_{2}$
This law also states that the force decreases with the square of the distance between them.
$F \propto \frac{1}{r^{2}}$
We can combine these into one equation that will help is analyze and solve the movement of celestial bodies anywhere in the Universe.
$F=G\frac{M_{1}M_{2}}{r^{2}}$
where $$G$$ is the gravitational constant. This law was deduced by Newton from observations and was formulated in the Principia.

If we were to look at the Sun and Earth system, we would have what is known as the two-body problem. While we may think this is a simple problem but in reality it isn’t. The Earth does not revolve around the Sun as if the more massive Sun was a stationary body but rather both the Earth and Sun revolve around a common point — the center of mass or barycenter. Though makes the problem a little more difficult, as this point is very close to the center of the Sun and we can make an approximation where the Sun is in the center and doesn’t move while the Earth moves around it.

If we use this technique, we can derive an equation that describes the Earth’s motion around the Sun. This is a classic “two-body” problem and its solutions describe the familiar elliptical orbits of the planets known since the time of Kepler. Unfortunately, when we add a third body to our equations of motion, such as a spacecraft lost in space between the Earth and the sun, we can no longer find an analytical solution; the equations become unsolvable.

While the three-body problem has no analytical solution, Joseph-Louis Lagrange in 1772 showed that if we restricted the mass of the third body to be so small that it couldn’t affect the other two, there were some solutions to be found. The solutions to this “restricted three-body problem” finds that the three bodies move in unison and always maintain the same position relative to each other.

These five points, where the third body stays in the same relative point in space, are known as Lagrange points and mark the positions where the combined gravitational pull of the two large masses precisely provides the force to orbit with them. They are labelled L1, L2, L3, L4 and L5 and also known as L-points or libration points.

## The Trojan Asteroids

The L4 and L5 libration points lie on the corners of two equilateral triangles. The points are balanced because the gravitational forces between the Earth and Sun keep any object in orbital equilibrium with the rest of the system. These points are sometimes referred to as “triangular Lagrange points” or “Trojan points” and come from the Trojan asteroids at the Sun-Jupiter L4 and L5 points. These asteroids were named after characters from Homer’s Iliad. Asteroids at the L4 point leads Jupiter and is referred to as the “Greek camp” while those are the L5 are referred to as the “Trojan camp”.

The Earth-Sun system, like the Jupiter-Sun system, also uses the same terms of reference. There is one known Greek asteroid in the Earth-Sun system, 2010 TK7, first detected in October 2010 by Dr. Martin Connors using the Wide-field Infrared Survey Explorer (WISE) . The asteroid’s designation come from the provisional naming system for minor planets. This asteroid which has a diameter of about 300 meters oscillates about the L4 Lagrangian point. The region around these points is known to contain interplanetary dust. As people tend to search for asteroids at much greater elongations, very few searches have been done at these locations. Currently, there are no confirmed or suspected L5 Trojans of Earth which makes both Raj’s and Alex’s work interesting and something to talk about.

## The Big Bang Theory of the Doppler Effect

Season 01, Episode 06: “The Middle-Earth Paradigm”

Sheldon Cooper dresses as the Doppler Effect on the Season 1, Episode 6 “The Middle-Earth Paradigm” of the Big Bang Theory

Yes. It’s the apparent change in the frequency of a wave caused by relative motion between the source of the wave and the observer.

-Sheldon Cooper

In the “Middle-Earth Paradigm” episode, Sheldon Cooper dresses as the “Doppler Effect” for Penny’s Halloween party. The Doppler Effect (or Doppler Shift) describes the change in pitch or frequency that results as a source of sound moves relative to an observer; moving relative can mean either the source is moving while the observer is stationary or vice versa. It is commonly heard when a siren approaches and recedes from an observer. As the siren approaches, the pitch sounds higher and lowers as it moves away. This effect was first proposed by Austrian physicist Christian Doppler in 1842 to explain the color of binary stars.

In 1845, Christophorus Henricus Diedericus (C. H. D.) Buys-Ballot, a Dutch chemist and meteorologist conducted the famous experiment to prove this effect. He assembled a group of horn players on an open cart attached to a locomotive. Ballot then instructed the engineer to rush past him as fast as he could while the musicians played and held a constant note. As the train approached and receded, Ballot noted that the pitch changed as he stood and listened on the stationary platform.

# Physics of the Doppler Effect

A stationary sound source has sound waves radiating outward and can be viewed as concentric circles.

As a stationary sound source produces sound waves, its wave-fronts propagate away from the source at a constant speed, the speed of sound. This can be seen as concentric circles moving away from the center. All observers will hear the same frequency, the frequency of the source of the sound.

When either the source or the observer moves relative to each other, the frequency of the sound that the source emits does not change but rather the observer hears a change in pitch. We can think of the following way. If a pitcher throws balls to someone across a field at a constant rate of one ball a second, the person will catch those balls at the same rate (one ball a second). Now if the pitcher runs towards the catcher, the catcher will catch the balls faster than one ball per second. This happens because as the catcher moves forward, he closes in the distance between himself and the catcher. When the pitcher tosses the next ball it has to travel a shorter distance and thus travels a shorter time. The opposite is true if the pitcher was to move away from the catcher.

If instead of the pitcher moving towards the catcher, the pitcher stayed stationary and the catcher ran forward. As the catcher runs forward, he closes in the distance between him and the pitcher so the time it takes from the ball to leave the pitcher’s hand to the catcher’s mitt is decreased. In this case, it also means that the catcher will catch the balls at a faster rate than the pitcher tosses them.

## Sub Sonic Speeds

The source radiates sound waves outward. As it moves, the center of each new wavefront is slightly displaced to the right and the wavefronts bunch up on the right side (front) and spread out further apart on the left side (behind) the source.

We can apply the same idea of the pitcher and catcher to a moving source of sound and an observer. As the source moves, it emits sounds waves which spread out radially around the source. As it moves forward, the wave-fronts in front of the source bunch up and the observer hears an increase in pitch. Behind the source, the wave-fronts spread apart and the observer standing behind hears a decrease in pitch.

### The Doppler Equation

When the speeds of source and the receiver relative to the medium (air) are lower than the velocity of sound in the medium, i.e. moves at sub-sonic speeds, we can define a relationship between the observed frequency, $$f$$, and the frequency emitted by the source, $$f_0$$.
$f = f_{0}\left(\frac{v + v_{o}}{v + v_{s}}\right)$
where $$v$$ is the speed of sound, $$v_{o}$$ is the velocity of the observer (this is positive if the observer is moving towards the source of sound) and $$v_{s}$$ is the velocity of the source (this is positive if the source is moving away from the observer).

#### Source Moving, Observer Stationary

We can now use the above equation to determine how the pitch changes as the source of sound moves towards the observer. i.e. $$v_{o} = 0$$.
$f = f_{0}\left(\frac{v}{v – v_{s}}\right)$
$$v_{s}$$ is negative because it is moving towards the observer and $$v – v_{s} < v$$. This makes $$v/(v - v_{s})$$ larger than 1 which means the pitch increases.

#### Source Stationary, Observer Moving

Now if the source of sound is still and the observer moves towards the sound, we get:
$f = f_{0}\left( \frac{v + v_{o}}{v} \right)$
$$v_{o}$$ is positive as it moves towards the source. The numerator is larger than the denominator which means that $$v + v_{o}/v$$ is greater than 1. The pitch increases.

## Speed of Sound

As the source of sound moves at the speed of sound the wave fronts in front of the source all bunch up at the same point.

As the source of sound moves at the speed of sound, the wave fronts in front become bunched up at the same point. The observer in front won’t hear anything until the source arrives. When the source arrives, the pressure front will be very intense and won’t be heard as a change in pitch but as a large “thump”.

The observer behind will hear a lower pitch as the source passes by.
$f = f_{0}\left( \frac{v – 0}{v + v} \right) = 0.5 f_{0}$

Early jet pilots flying at the speed of sound (Mach 1) reported a noticeable “wall” or “barrier” had to be penetrated before achieving supersonic speeds. This “wall” is due to the intense pressure front, and flying within this pressure front produced a very turbulent and bouncy ride. Chuck Yeager was the first person to break the sound barrier when he flew faster than the speed of sound in the Bell X-1 rocket-powered aircraft on October 14, 1947.

Bell X-1 rocket plane of the United States Air Force (NASA photography)

As the science of super-sonic flight became better understood, engineers made a number changes to aircraft design that led the the disappearance of the “sound barrier”. Aircraft wings were swept back and engine performance increased. By the 1950s combat aircraft could routinely break the sound barrier.

## Super-Sonic

As the source moves faster than the speed of sound, i.e. faster than the sound waves it creates, it leads its own advancing wavefront. The sound source will pass by the stationary observer before the observer hears the sound.

As the sound source breaks and moves past the “sound barrier”, the source now moves faster than the sound waves it creates and leads the advancing wavefront. The source will pass the observer before the observer hears the sound it creates. As the source moves forward, it creates a Mach cone. The intense preseure front on the Mach cone creates a shock wave known as a “sonic boom”.

### Twice the Speed of Sound

Something interesting happens when the source moves towards the observer at twice the speed of sound — the tone becomes time reversed. If music was being played, the observer will hear the piece with the correct tone but played backwards. This was first predicted by Lord Rayleigh in 1896 .

We can see this by using the Doppler Equation.
$f = f_{0}\left(\frac{v}{v-2v}\right)$
This reduces to
$f=-f_{0}$
which is negative because the sound is time reversed or is heard backwards.

# Applications

U.S. Army soldier uses a radar speed gun to catch speeding violators at Tallil Air Base, Iraq.

The Doppler Effect is used in radar guns to measure the speed of motorists. A radar beam is fired at a moving target as it approaches or recedes from the radar source. The moving target then reflects the Doppler-shifted radar wave back to the detector and the frequency shift measured and the motorist’s speed calculated.

We can combine both cases of the Doppler equation to give us the relationship between the reflected frequency ($$f_{r}$$) and the source frequency ($$f$$):
$f_{r} = f \left(\frac{c+v}{c-v}\right)$
where $$c$$ is the speed of light and $$v$$ is the speed of the moving vehicle. The difference between the reflected frequency and the source frequency is too small to be measured accurately so the radar gun uses a special trick that is familiar to musicians – interference beats.

To tune a piano, the pitch can be adjusted by changing the tension on the strings. By using a tuning instrument (such as a tuning fork) which can produce a sustained tone over time, a beat frequency can be heard when placed next to the vibrating piano wire. The beat frequency is an interference between two sounds with slightly different frequencies and can be herd as a periodic change in volume over time. This frequency tells us how far off the piano strings are compared to the reference (tuning fork).

To detect this change in a radar gun does something similar. The returning wave is “mixed” with the transmitted signal to create a beat note. This beat signal or “heterodyne” is then measured and the speed of the vehicle calculated. The change in frequency or the difference between $$f_{r}$$ and $$f$$ or $$\Delta f$$ is
$f_{r} – f = f\frac{2v}{c-v}$
as the difference between the speed of light, $$c$$, and the speed of the vehicle, $$v$$, is small, we can approximate this to
$\Delta f \approx f\frac{2v}{c}$
By measuring this frequency shift or beat frequency, the radar gun can calculate and display a vehicle’s speed.

# “I am the Doppler Effect”

The Doppler Effect is an important principle in physics and is used in astronomy to measure the speeds at which galaxies and stars are approaching or receding from us. It is also used in plasma physics to estimate the temperature of plasmas. Plasmas are one of the four fundamental states of matter (the others being solid, liquid, and gas) and is made up of very hot, ionized gases. Their composition can be determined by the spectral lines they emit. As each particle jostles about, the light emitted by each particle is Doppler shifted and is seen as a broadened spectral line. This line shape is called a Doppler profile and the width of the line is proportional to the square root of the temperature of plasma gas. By measuring the width, scientists can infer the gas’ temperature.

We can now understand Sheldon’s fascination with the Doppler Effect as he aptly explains and demonstrates its effects. As an emergency vehicle approaches an observer, its siren will start out with a higher pitch and slide down as as it passes and moves away from the observer. This can be heard as the (confusing) sound he demonstrates to Penny’s confused guests.

# References

## The Big Bang Theory of the Inclined Plane

Season 01, Episode 02: “The Big Bran Hypothesis”

Sheldon and Leonard contemplate using the stairs as an inclined plane to move Penny’s furniture to her apartment.

In Season 1 Episode 2 of The Big Bang Theory, “The Big Bran Hypothesis”, Penny (Kaley Cuoco) asks Leonard (Johnny Galecki) to sign for a furniture delivery if she isn’t home. Unfortunately for Leonard and Sheldon, they are left with the task of getting a huge (and heavy) box up to Penny’s apartment.

To solve this problem, Leonard suggest using the stairs as an inclined plane, one of the six classical simple machines defined by Renaissance scientists. Both Leonard and Sheldon have the right idea here. Not only are inclined planes used to raise heavy loads but they require less effort to do so. Though this may make moving a heavy load easier the tradeoff is that the load must now be moved over a greater distance. So while, as Leonard correctly calculates, the effort required to move Penny’s furniture is reduced by half, the distance he and Sheldon must move Penny’s furniture twice the distance to raise it directly.

# Mathematics of the Inclined Plane

## Effort to lift block on Inclined Plane

Now we got an inclined plane. Force required to lift is reduced by the sine of the angle of the stairs… call it 30 degrees, so about half.

Free-body Diagram of a block on an inclined plane. It shows the forces acting on a block and the force needed to keep it stationary and not let it slip down.

To analyze the forces acting on a body, physicists and engineers use rough sketches or free body diagrams. This diagram can help physicists model a problem on paper and to determine how forces act on an object. We can resolve the forces to see the effort needed to move the block up the stairs.

If the weight of Penny’s furniture is $$W$$ and the angle of the stairs is $$\theta$$ then
$\angle_{\mathrm{stairs}}\equiv\theta \approx 30^\circ$
and
$\Rightarrow\sin 30^\circ = \frac{1}{2}$
So the effort needed to keep the box in place is about half the weight of the furniture box or $$\frac{1}{2}W$$, just as Leonard says.

## Distance moved along Inclined Plane

The relationship between the height $$h$$ the block is raised and the distance it moves $$d$$ is the sine of the angle $$\theta$$.

While the inclined plane allows Leonard and Sheldon to push the box with less effort, the tradeoff is that the distance they move along the incline is twice the height to raise the box vertically. Geometry shows us that
$\sin \theta = \frac{h}{d}$
We again assume that the angle of the stairs is approximately $$30^\circ$$ and $$\sin 30^{\circ} = 1/2$$ then we have $$d=2h$$.

# Uses of the Inclined Plane

We see inclined planes daily without realizing it. They are used as loading ramps to load and unload goods. Wheelchair ramps also allow wheelchair users, as well as users of strollers and carts, to access buildings easily. Roads sometimes have inclined planes to form a gradual slope to allow vehicles to move over hills without losing traction. Inclined planes have also played an important part in history and were used to build the Egyptian pyramids and possibly used to move the heavy stones to build Stonehenge.

## Lombard Street (San Francisco)

Lombard Street is one of the most visited street in San Francisco as seen from Coit Tower. It is best known for the one-way section on Russian Hill between Hyde and Leavenworth Streets, in which the roadway has eight sharp turns (or switchbacks) that have earned the street the distinction of being the crookedest “most winding “street in the world, though this title is contested. (Photo by David Yu).

Lombard Street in San Francisco is famous for its eight tight hairpin turns (or switchbacks) that have earned it the distinction of being the crookedest street in the world (though this title is contested). These eight switchbacks are crucial to the street’s design as the reduce the hills natural 27° grade which is too steep for most vehicles. It is also a hazard to pedestrians, who are more accustomed to a more reasonable 4.86° incline due to wheel chair navigability concerns.

Technically speaking, the “zigzag” path doesn’t make climbing or coming down the hill any easier. As we have seen, all it does is change how various forces are applied. It just requires less effort to move up or down but the tradeoff is that you travel a longer distance. This has several advantages. Car engines have to be less powerful to climb the hill and in the case of descent, less force needs to be applied on the brakes. There are also safety considerations. A car will not accelerate down the switch back path as fast than if it was driven straight down, making speeds safer and more manageable for motorists.

This idea of using zigzagging paths to climb steep hills and mountains is also used by hikers and rock climbers for very much the same reason Lombard Street zigszags. The tradeoff is that the distance traveled along the path is greater than if a climber goes straight up.

# The Descendants of Archimedes

We don’t need strength, we’re physicists. We are the intellectual descendants of Archimedes. Give me a fulcrum and a lever and I can move the Earth. It’s just a matter of… I don’t have this, I don’t have this!

We see that Leonard had the right idea. If we were to assume are to assume — based on the size of the box — that the furniture is approximately 150 lbs (65kg) and the effort is reduced by half, then they need to push with at least 75 lbs of force. This is equivalent to moving a 34kg mass. If they both push equally, they are each left pushing a very manageable 37.5 lbs, the equivalent of pushing a 17kg mass.

Penny’s apartment is on the fourth floor and we if we assume a standard US building design of ten feet per floor, this means a 30 foot vertical rise. The boys are left with the choice of lifting 150 lbs vertically 30 feet or moving 75lbs a distance of 60 feet. The latter is more manageable but then again, neither of our heroes have any upper body strength.