Tag: Mathematics

School bus routes are expensive and hard to plan. We calculated a better way

Here’s a math problem even the brightest school districts struggle to solve: getting hordes of elementary, middle and high school students onto buses and to school on time every day. The Conversation

Transporting all of these pupils presents a large and complex problem. Some school districts use existing software systems to develop their bus routes. Others still develop these routes manually.

In such problems, improving operational efficiency even a little could result in great advantages. Each school bus costs school districts somewhere between US$60,000 and $100,000. So, scheduling the buses more efficiently will result in significant monetary savings.

Over the past year, we have been working with the Howard County Public School System (HCPSS) in Maryland to analyze its transportation system and recommend ways to improve it. We have developed a way to optimize school bus routes, thanks to new mathematical models.

Finding the optimal solution to this problem is very valuable, even if that optimal solution is only slightly better than the current plan. A solution that is only one percent worse would require a considerable number of additional buses due to the size of the operation.

By optimizing bus routes, schools can cut down on costs, while still serving all of the children in their district. Our analysis shows that HCPSS can save between five and seven percent on the number of buses needed.

Route planning

A bus trip in the afternoon starts from a given school and visits a sequence of stops, dropping off students until the bus is empty. A route is a sequence of trips from different schools that are linked together to be served by one bus.

Our goal was to reduce both the total time buses run without students on board – also known as aggregate deadhead time – as well as the number of routes. Fewer routes require fewer buses since each route is assigned to a single bus. Our approach uses data analysis and mathematical modeling to find the optimal solution in a relatively short time.

To solve this problem, a computer algorithm considers all of the bus trips in the district. Without modifying the trips, the algorithm assigns them to routes such that the aggregate deadhead time and the number of routes are minimized. Individual routes become longer, allowing the bus to serve more trips in a single route.

Since the trips are fixed, in this way we can decrease the total time the buses are en route. Minimizing the deadhead travel results in cost savings and reductions in air pollution.

The routes that we generated can be viewed as a lower bound to the number of buses needed by school districts. We can find the optimal solution for HCPSS in less than a minute.

Serving all students

While we were working on routes, we decided to also tackle the problem of the bus trips themselves. To do this, we needed to determine what trips are required to serve the students for each school in the system, given bus capacities, stop locations and the number of students at each stop. This has a direct impact on how routes are chosen.

Most existing models aim to minimize either the total travel time or the total number of trips. The belief in such cases is that, by minimizing the number of trips, you can minimize the number of buses needed overall.

However, our work shows that this is not always the case. We found a way to cut down on the number of buses needed to satisfy transportation demands, without trying to minimize either of the above two objectives. Our approach considers not only minimizing the number of trips but also how these trips can be linked together.

New start times

Last October, we presented our work at the Maryland Association of Pupil Transportation conference. An audience member at that conference suggested that we analyze school start and dismissal times. By changing the high school, middle school and elementary school start times, bus operations could potentially be even more efficient. Slight changes in school start times can make it possible to link more trips together in a single bus route, hence decreasing the number of buses needed overall.

We developed a model that optimizes the school bell times, given that each of the elementary, middle and high school start times fall within a prespecified time window. For example, the time window for elementary school start times would be from 8:15 to 9:25 a.m.; for middle schools, from 7:40 to 8:30 a.m.; and all high schools would start at 7:25 a.m.

Our model looks at all of the bus trips and searches for the optimal combination of school dismissal time such that the number of school buses, which is the major contributing factor to costs, is minimized. We found that, in most cases, optimizing the bell times results in significant savings regarding the number of buses.

Next steps

Using our model, we ran many different “what if?” scenarios using different school start and dismissal times for the HCPSS. Four of these are currently under consideration by the Howard County School Board for possible implementation.

We are also continuing to enhance our current school bus transportation models, as well developing new ways to further improve efficiency and reduce costs.

For example, we are building models that can help schools select the right vendors for their transportation needs, as well as minimize the number of hours that buses run per day.

In the future, the type of models we are working on could be bundled into a software system that schools can use by themselves. There is really no impediment in using these types of systems as long as the school systems have an electronic database of their stops, trips, and routes.

Such software could potentially be implemented in all school districts in the nation. Many of these districts would benefit from using such models to evaluate their current operations and determine if any savings can be realized. With many municipalities struggling with budgets, this sort of innovation could save money without degrading service.

Ali Haghani, Professor of Civil & Environmental Engineering, University of Maryland and Ali Shafahi, Ph.D. Candidate in Computer Science, University of Maryland

Photo Credit: Dean Hochman


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This article was originally published on The Conversation. Read the original article.

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3.14 essential reads about π for Pi Day

Editor’s note: The following is a roundup of archival stories. The Conversation

On March 14, or 3/14, mathematicians and other obscure-holiday aficionados celebrate Pi Day, honoring π, the Greek symbol representing an irrational number that begins with 3.14. Pi, as schoolteachers everywhere repeat, represents the ratio of a circle’s circumference to its diameter.

What is Pi Day, and what, really, do we know about π anyway? Here are three-and-bit-more articles to round out your Pi Day festivities.

A silly holiday

First off, a reflection on this “holiday” construct. Pi itself is very important, writes mathematics professor Daniel Ullman of George Washington University, but celebrating it is absurd:

The Gregorian calendar, the decimal system, the Greek alphabet, and pies are relatively modern, human-made inventions, chosen arbitrarily among many equivalent choices. Of course a mood-boosting piece of lemon meringue could be just what many math lovers need in the middle of March at the end of a long winter. But there’s an element of absurdity to celebrating π by noting its connections with these ephemera, which have themselves no connection to π at all, just as absurd as it would be to celebrate Earth Day by eating foods that start with the letter “E.”

And yet, here we are, looking at the calendar and getting goofily giddy about the sequence of numbers it shows us.

There’s never enough

In fact, as Jon Borwein of the University of Newcastle and David H. Bailey of the University of California, Davis, document, π is having a sustained cultural moment, popping up in literature, film, and song:

Sometimes the attention given to pi is annoying. On 14 August 2012, the U.S. Census Office announced the population of the country had passed exactly 314,159,265. Such precision was, of course, completely unwarranted. But sometimes the attention is breathtakingly pleasurable.

Come to think of it, pi can indeed be a source of great pleasure. Apple’s always comforting, and cherry packs a tart pop. Chocolate cream, though, might just be where it’s at.

Strange connections

Of course, π appears in all kinds of places that relate to circles. But it crops up in other places, too – often where circles are hiding in plain sight. Lorenzo Sadun, a professor of mathematics at the University of Texas at Austin, explores surprising appearances:

Pi also crops up in probability. The function f(x)=e-x², where e=2.71828… is Euler’s number, describes the most common probability distribution seen in the real world, governing everything from SAT scores to locations of darts thrown at a target. The area under this curve is exactly the square root of π.

It’s enough to make your head spin.

Historical pi

If you want to engage with π more directly, follow the lead of Georgia State University mathematician Xiaojing Ye, whose guide starts thousands of years ago:

The earliest written approximations of pi are 3.125 in Babylon (1900-1600 B.C.) and 3.1605 in ancient Egypt (1650 B.C.). Both approximations start with 3.1 – pretty close to the actual value, but still relatively far off.

By the end of his article, you’ll find a method to calculate π for yourself. You can even try it at home!

An irrational bonus

And because π is irrational, we’ll irrationally give you even one more, from education professor Gareth Ffowc Roberts at Bangor University in Wales, who highlights the very humble beginnings of the symbol π:

After attending a charity school, William Jones of the parish of Llanfihangel Tre’r Beirdd landed a job as a merchant’s accountant and then as a maths teacher on a warship, before publishing A New Compendium of the Whole Art of Navigation, his first book in 1702 on the mathematics of navigation. On his return to Britain he began to teach maths in London, possibly starting by holding classes in coffee shops for a small fee.

Shortly afterwards he published “Synopsis palmariorum matheseos,” a summary of the current state of the art developments in mathematics which reflected his own particular interests. In it is the first recorded use of the symbol π as the number that gives the ratio of a circle’s circumference to its diameter.

What made him realize that this ratio needed a symbol to represent a numeric value? And why did he choose π? It’s all Greek to us.

Jeff Inglis, Editor, Science + Technology, The Conversation

Photo Credit: Yelp Inc.

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This article was originally published on The Conversation. Read the original article.

Episode II: Math Jedi Matt Joins Black Jedi Don and Jamar on the BRBD Show

By TSS Admin

This Saturday evening, March 11th, at 6 pm central standard time on Twin Cities News Talk, Matt Johnson, our Editor-in-chief and mathematician, will be making his second guest apprearance on the Black Republican/Black Democrat show (BRBD).

Here’s the link to Matt’s first appearance on BRBD:

He will be joining co-hosts Donald Allen (R) and Jamar Nelson (D), and roving reporter Preya Samsundar from Alpha News, on the Black Jedi Radio Network to discuss Minneapolis economics and politics, why the presidential election polls and forecasts weren’t wrong, and Bill Nye “The Science Guy” and Tucker Carlson’s now infamous climate change exchange. This is sure to be a light-saber blazing event with a large audience.

Speaking of a large audience, the Black Republican/Black Democrat show has blown up on social media since Matt’s last visit on February 11th of this year. During Matt’s first visit, BRBD had 1,535 followers on their Facebook page. Since then, the Black Jedi Radio Network has gained nearly 5,000 followers; and so this time around, the Math Jedi Matt Johnson will have a much larger audience to share the gospel of mathematics with, while dueling with republicans and democrats.

Where can you listen?

For our Twin Cities’ readers, just simply turn the terrestrial dial to AM 1130 or FM 103.5. For our national readers, just download the iHeartRadio app or you can listen LIVE via the world-wide web by going to www.TwinCitiesNewsTalk.com, which is an iHeartRadio station. For our readers who would like to call into the show, dial (612) 986 – 0010.

We’ll see you Saturday night!

 

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Mathematics is beautiful (no, really)

For many people, memories of maths lessons at school are anything but pretty. Yet “beautiful” is a word that I and other mathematicians often use to describe our subject. How on earth can maths be beautiful – and does it matter?

For me, as a mathematician, it is hugely important. My enjoyment of the beauty of mathematics is part of what motivates me to study the subject. It is also a guide when I am working on a problem: if I think of a few strategies, I will choose the one that seems most elegant first. And if my solution seems clumsy then I will revisit it to try to make it more attractive.

I’ve just finished marking a pile of homework from my second-year mathematics undergraduates. I am struck by two students’ contrasting solutions to one problem. Both solutions are correct, both answer the question. And yet I much prefer one to the other. It’s not just that one is longer than the other, or that one is explained better than the other (both are described well, in fact).

The longer one doesn’t quite get to the heart of the matter, it’s a bit cluttered with unnecessary distractions. The other uses a different approach, which captures the essence of the ideas – it helps the reader to understand why this piece of mathematics works this way, not just that it does. For a mathematician, the “why” is critical, and we are always looking for arguments that reveal this.

Some cases of mathematical beauty are clear. Fractals, for example, are mathematical sets of numbers – corresponding to shapes – that have striking self-similarity and that have inspired numerous artists.

Less is more

But what about less obvious cases? Let me try to give you an example. Perhaps you recognise the sequence of numbers 1, 3, 6, 10, 15, 21, 28, … This is a sequence that students often encounter at school: the triangular numbers. Each number in the sequence corresponds to the number of dots in a sequence of triangles.

The six first triangular numbers: 1, 3, 6, 10, 15, 21.

Can we predict what the 1000th number in the sequence will be? There are many ways to tackle this question, and in fact unpicking the similarities and differences between these approaches is in itself both mathematical and enlightening. But here is one rather beautiful argument.

Imagine the 10th number in the sequence (because it’s easier to draw the picture than for the 1000th!). Let’s count the dots without counting the dots. We have a triangle of dots, with 10 in the bottom row and 10 rows of dots.

If we make another copy of that arrangement, we can rotate it and put it next to our original triangle of dots – so that the two triangles form a rectangle. This shape of dots will have 10 in the bottom row and 11 rows, so there are 10 x 11 = 110 dots in total (see figure below). Now we know that half of those were in our original triangle, so the 10th triangular number is 110/2 = 55. And we didn’t have to count them.

The 10th triangular number x2.

The power of this mathematical argument is that we can painlessly generalise to any number – even without drawing the dots. We can do a thought experiment. The 1000th triangle in the sequence will have 1000 dots in the bottom row, and 1000 rows of dots. By making another copy of this and rotating it, we get a rectangle with 1000 dots in the bottom row and 1001 rows. Half of those dots were in the original triangle, so the 1000th triangular number is (1000 x 1001)/2 = 500500.

For me, this idea of drawing the dots, duplicating, rotating and making a rectangle is beautiful. The argument is powerful, it generalises neatly (to any size of a triangle), and it reveals why the answer is what it is.

There are other ways to predict this number. One is to look at the first few terms of the sequence, guess a formula, and then prove that the formula does work (for example by using a technique called proof by induction). But that doesn’t convey the same memorable explanation behind the formula. There is an economy to the argument with pictures of dots, a single diagram captures everything we need to know.

Here’s another argument that I find attractive. Let’s think about the sum below:

The harmonic series.

This is the famous harmonic series. It turns out that it doesn’t equal a finite number – mathematicians say that the sum “diverges”. How can we prove that? It sounds difficult, but one elegant idea does the job.

The harmonic series with grouped terms.

Here each group of fractions adds up to more than ½. We know that ⅓ is bigger than ¼. That means (⅓) + (¼) is bigger than (¼) + (¼), which equals ½. So by adding enough blocks, each bigger than ½, the sum gets bigger and bigger – we can beat any target we like. By adding an infinite number of them we will get an infinite sum. We have tamed the infinite, with a beautiful argument.

A waiting game?

These are not the most difficult pieces of mathematics. One of the challenges of mathematics is that tackling more sophisticated problems often means first tackling more sophisticated terminology and notation. I cannot find a piece of mathematics beautiful unless I first understand it properly – and that means it can take a while for me to appreciate the aesthetic qualities.

I don’t think this unique to mathematics. There are pieces of music, buildings, pieces of visual art where I have not at first appreciated their beauty or elegance – and it is only by persevering, by grappling with the ideas, that I have come to perceive the beauty.

For me, one of the joys of teaching undergraduates is watching them develop their own appreciation of the beauty of mathematics. I’m going to see my second years this afternoon to go over their homework, and I already know that we’re going to have an interesting conversation about their different solutions – and that considering the aesthetic qualities will play a part in deepening their understanding of the mathematics.

School students can have just the same experience: when they’re given the opportunity to engage with rich questions, when they can play with mathematical ideas, when they have the chance to experience multiple strategies to the same question rather than just getting the answer in the back of the textbook and moving on. The mathematical ideas do not have to be university level, there are beautiful problems that are perfect for school students. Happily, there are many maths teachers and maths education projects that are helping students to have those experiences of the beauty of mathematics.

The Conversation

Vicky Neale, Whitehead Lecturer at the Mathematical Institute and Supernumerary Fellow at Balliol College, University of Oxford

Photo Credit: Ankush Sabharwal – CC BY-SA

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This article was originally published on The Conversation. Read the original article.

Introducing the terrifying mathematics of the Anthropocene

Here are some surprising facts about humans’ effect on planet Earth. We have made enough concrete to create an exact replica of Earth 2mm thick. We have produced enough plastic to wrap Earth in clingfilm. We are creating “technofossils”, a new term for congealed human-made materials – plastics and concretes – that will be around for tens of millions of years.

But it is the scale that humans have altered Earth’s life support system that is the most concerning.

In 2000, Nobel laureate Paul Crutzen and Eugene Stoermer proposed that human impact on the atmosphere, the oceans, the land and ice sheets had reached such a scale that it had pushed Earth into a new epoch. They called it the Anthropocene and argued the current Holocene epoch was over.

The Holocene began 11,700 years ago as we emerged from a deep ice age. Over the past 10,000 years, the defining feature of the Holocene has been a remarkably stable Earth system. This stability has allowed us to develop agriculture and hence villages, towns and eventually cities – human civilization.

We use pretty powerful rhetoric to describe the Anthropocene and current human impact. As The Economist stated in 2011, humanity has “become a force of nature reshaping the planet on a geological scale”. We are like an asteroid strike. We have the impact of an ice age.

But what does this really mean? Does it mean, for example, that we are having as big an impact as these natural forces are having right now, or is it, somehow, more profound?


Humans: the new asteroids.
Steve Jurvetson, CC BY

The maths of the Anthropocene

In our recent study, we wanted to find the simplest way to mathematically describe the Anthropocene and articulate the difference between how the planet once functioned and how it now functions.

Life on Earth, the chemical and physical composition of the atmosphere and oceans, and the size of the ice sheets have changed over time because of slight alterations to Earth’s orbit around the sun, changes to the sun’s energy output or major asteroid impacts like the one that killed the dinosaurs.

Cyanobacteria changed the world; now it’s our turn.
Matthew J Parker, CC BY-SA

They can also change due to geophysical forces: continents collide, cutting off ocean currents so heat is distributed in a new way, upsetting climate and biodiversity.

They also shift due to sheer internal dynamics of the system – new life evolves to drive great planetary shifts, such as the Great Oxidation Event around 2.5 billion years ago when newly evolved cyanobacteria began emitting the deadly poison oxygen that killed all simple life forms it came in touch with. Life had to evolve to tolerate oxygen.

Taking as our starting point a 1999 article by Earth system scientist Hans Joachim Schellnhuber, we can say the rate of change of the Earth system (E) has been driven by three things: astronomical forcings such as those from the sun or asteroids; geophysical forcing, for example changing currents; and internal dynamics, such as the evolution of cyanobacteria. Let’s call them A, G and I.

Mathematically, we can put it like this:

It reads: the rate of change of the Earth system (dE/dt) is a function of astronomical and geophysical forcings and internal dynamics. It is a very simple statement about the main drivers of the system.

This equation has been true for four billion years since the first life evolved. In his article, Schellnhuber argued that people must be added into this mix, but his theory came before the full impact of humanity had been assessed. In the past few decades, this equation has been radically altered.

We are losing biodiversity at rates tens to hundreds of times faster than natural rates. Indeed, we are approaching mass extinction rates. There have been five mass extinctions in the history of life on Earth. The last killed the non-avian dinosaurs 66 million years ago, now humans are causing the sixth.

The rate we are emitting carbon dioxide might be at an all-time high since that time too. Global temperatures are rising at a rate 170 times faster than the Holocene baseline. The global nitrogen cycle is undergoing its largest and most rapid change in possibly 2.5 billion years.

In fact, the rate of change of the Earth system under the human influence in the past four decades is so significant we can now show that the equation has become:

H stands for humanity. In the Anthropocene Equation, the rate of change of the Earth system is a function of humanity.

A, G and I are now approaching zero relative to the other big force – us – they have become essentially negligible. We are now the dominant influence on the stability and resilience of the planet we call home.

This is worth a little reflection. For four billion years, the Earth system changed under the influence of tremendous solar-system wide forces of nature. Now, this no longer holds.


IPCC, 2014: Climate Change 2014: Synthesis Report. Contribution of Working Groups I, II and III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. IPCC, Geneva, Switzerland

A new reality

Heavenly bodies of course still exert some force; so does the ground beneath our feet. But the rates at which these forces operate are now negligible compared with the rate at which we are changing the Earth system. In the 1950s or 1960s, our own impact rivaled the great forces of nature. Now it usurps them entirely.

This should come as a shock not only to environmentalists but to everyone on Earth. But our conclusion is arguably a modest addition to the canon of academic literature. The scale and rate of change have already been well established by Earth system scientists over the past two decades.

Recently, Mark Williams and colleagues argued that the Anthropocene represents the third new era in Earth’s biosphere, and astrobiologist David Grinspoon argued that the Anthropocene marks one of the major events in a planet’s “life”, when self-aware cognitive processes become a key part of the way the planet functions.

Still, formalizing the Anthropocene mathematically brings home an entirely new reality.

The drama is heightened when we consider that for much of Earth’s history the planet has been either very hot – a greenhouse world – or very cold – an icehouse world. These appear to be the deeply stable states lasting millions of years and resistant to even quite major shoves from astronomical or geophysical forces.

But the past 2.5 million years have been uncharacteristically unstable, periodically flickering from cold to a gentle warmth.

The consumption vortex

So, who do we mean when we talk of H? Some will argue that we cannot treat humanity as one homogenous whole. We agree.

While all of humanity is now in the Anthropocene, we are not all in it in the same way. Industrialized societies are the reason we have arrived at this place, not Inuits in northern Canada or smallholder farmers in sub-Saharan Africa.

Scientific and technological innovations and economic policies promoting growth at all costs have created a consumption and production vortex on a collision course with the Earth system.

Others may say that natural forces are too important to ignore; for example, the El Niño weather system periodically changes patterns globally and causes Earth to warm for a year or so, and the tides generate more energy than all of humanity. But a warm El Niño is balanced by a cool La Niña. The tides and other great forces of nature are powerful but stable. Overall, they do not affect the rate of change of the Earth system.

Now, only a truly catastrophic volcanic eruption or direct asteroid hit could match us for impact.

So, can the Anthropocene equation be solved? The current rate of change must return to around zero as soon as possible. It cannot continue indefinitely. Either humanity puts on the brakes or it would seem unlikely a global civilization will continue to function on a destabilized planet. The choice is ours.

The Conversation

Owen Gaffney, Anthropocene analyst and communicator. Co-founder Future Earth Media Lab, Director of media (Stockholm Resilience Centre), Stockholm University and Will Steffen, Adjunct Professor, Fenner School of Environment and Society, Australian National University

Photo Credit: Maxpixel

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This article was originally published on The Conversation. Read the original article.

Radio Jedi co-hosts Donald and Jamar invite TSS’s Matt Johnson onto the BRBD Show

By TSS Admin

brbd-v1Our very own Editor-in-chief, and research scientist, Matt Johnson will be making his radio debut as a guest on the Black Republican Black Democrat Show this Saturday, February 11th, at 6 pm on Twin Cities News Talk in Minneapolis, Minnesota.

He will join radio Jedi co-hosts Donald Allen (R) and Jamar Nelson (D) for the 6 to 7 pm central time hour. Together, they will take a closer look at the socio-economic data – crime, employment, housing, etc. – for Minneapolis, and other American cities. They will be delving into Matt’s “Number Shrewdness” to get the real scoop on the urban numbers that are not always presented in a truthful light.

What’s going on in Chicago and other cities? Why is there such disparity in economic wealth between racial groups? What might be done to address such issues? These are just a few of the questions that may be addressed during this Saturday’s show.

Where do you listen?

For our Twin Cities’ readers, just simply turn the terrestrial dial to AM 1130 or FM 103.5. For our national readers, just simply download the iHeartRadio app or you can listen LIVE via the world-wide web by going to www.TwinCitiesNewsTalk.com, which is an iHeartRadio station.

 

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Chandra Images Show That Geometry Solves a Pulsar Puzzle

NASA’S Chandra X-ray Observatory has taken deep exposures of two nearby energetic pulsars flying through the Milky Way galaxy. The shape of their X-ray emission suggests there is a geometrical explanation for puzzling differences in behavior shown by some pulsars.

Pulsars − rapidly rotating, highly magnetized, neutron stars born in supernova explosions triggered by the collapse of massive stars − were discovered 50 years ago via their pulsed, highly regular, radio emission.  Pulsars produce a lighthouse-like beam of radiation that astronomers detect as pulses as the pulsar’s rotation sweeps the beam across the sky.

Since their discovery, thousands of pulsars have been discovered, many of which produce beams of radio waves and gamma rays. Some pulsars show only radio pulses and others show only gamma-ray pulses. Chandra observations have revealed steadier X-ray emission from extensive clouds of high-energy particles, called pulsar wind nebulas, associated with both types of pulsars. New Chandra data on pulsar wind nebulas may explain the presence or absence of radio and gamma-ray pulses.

This four-panel graphic shows the two pulsars observed by Chandra. Geminga is in the upper left and B0355+54 is in the upper right. In both of these images, Chandra’s X-rays, colored blue and purple, are combined with infrared data from NASA’s Spitzer Space Telescope that shows stars in the field of view. Below each data image, an artist’s illustration depicts more details of what astronomers think the structure of each pulsar wind nebula looks like.

For Geminga, a deep Chandra observation totaling nearly eight days over several years was analyzed to show sweeping, arced trails spanning half a light year and a narrow structure directly behind the pulsar. A five-day Chandra observation of the second pulsar, B0355+54, showed a cap of emission followed by a narrow double trail extending almost five light years.

The underlying pulsars are quite similar, both rotating about five times per second and both aged about half a million years. However, Geminga shows gamma-ray pulses with no bright radio emission, while B0355+54 is one of the brightest radio pulsars known yet not seen in gamma rays.

A likely interpretation of the Chandra images is that the long narrow trails to the side of Geminga and the double tail of B0355+54 represent narrow jets emanating from the pulsar’s spin poles. Both pulsars also contain a torus of emission spreading from the pulsar’s spin equator. These disk-shaped structures and the jets are crushed and swept back as the pulsars fly through the Galaxy at supersonic speeds

In the case of Geminga, the view of the torus is close to edge-on, while the jets point out to the sides.  B0355+54 has a similar structure, but with the torus viewed nearly face-on and the jets pointing nearly directly towards and away from Earth. In B0355+54, the swept-back jets appear to lie almost on top of each other, giving a doubled tail.

Both pulsars have magnetic poles quite close to their spin poles, as is the case for the Earth’s magnetic field. These magnetic poles are the site of pulsar radio emission so astronomers expect the radio beams to point in a similar direction as the jets. By contrast the gamma-ray emission is mainly produced along the spin equator and so aligns with the torus.

For Geminga, astronomers view the bright gamma-ray pulses along the edge of the torus, but the radio beams near the jets point off to the sides and remain unseen. For B0355+54, a jet points almost along our line of sight towards the pulsar. This means astronomers see the bright radio pulses, while the torus and its associated gamma-ray emission are directed in a perpendicular direction to our line of sight, missing the Earth.

These two deep Chandra images have, therefore, exposed the spin orientation of these pulsars, helping to explain the presence, and absence, of the radio and gamma-ray pulses.

The Chandra observations of Geminga and B0355+54 are part of a large campaign, led by Roger Romani of Stanford University, to study six pulsars that have been seen to emit gamma-rays. The survey sample covers a range of ages, spin-down properties and expected inclinations, making it a powerful test of pulsar emission models.

A paper on Geminga led by Bettina Posselt of Penn State University was accepted for publication in The Astrophysical Journal and is available online. A paper on B0355+54 led by Noel Klingler of the George Washington University was published in the December 20th, 2016 issue of The Astrophysical Journal and is available online. NASA’s Marshall Space Flight Center in Huntsville, Alabama, manages the Chandra program for NASA’s Science Mission Directorate in Washington. The Smithsonian Astrophysical Observatory in Cambridge, Massachusetts, controls Chandra’s science and flight operations.

Image credit: Geminga image: NASA/CXC/PSU/B. Posselt et al; Infrared: NASA/JPL-Caltech; B0355+54: X-ray: NASA/CXC/GWU/N. Klingler et al; Infrared: NASA/JPL-Caltech; Illustrations: Nahks TrEhnl

Read More from NASA’s Chandra X-ray Observatory.

For more Chandra images, multimedia and related materials, visit:

http://www.nasa.gov/chandra

Editor: Lee Mohon
Photo Credit: NASA

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