text:
MPI for the Study of Religious and Ethnic Diversity
New insights into economic and social change
Researchers of the Max Planck Society and the University of Cambridge collaborate in the Max Planck - Cambridge Centre for Ethics, Economy and Social Change
International Social Sciences
How do moral and ethical convictions affect local economies and impact global capitalism? This is the fundamental question driving the projects of the Max Planck - Cambridge Centre for Ethics, Economy and Social Change, which was officially inaugurated this Tuesday. "Without any doubt, in today's globalized world, the dynamics between ethics, religion and economy have reached an unprecedented complexity. This makes the research of the new Max Planck - Cambridge Centre very relevant for our times. This cooperation brings together the complementary skills of outstanding scientists of the Max Planck Society and the University of Cambridge. I am delighted that, in this period of uncertainty caused by Brexit, we have established another highly visible collaboration with top British scientists," says Martin Stratmann, President of the Max Planck Society. He was together with Stephen Toope, Vice-Chancellor of the University of Cambridge, and the four Co-Directors among the speakers during the official launch ceremony.
Scene from a shopping mall. Postdoctoral fellows of the new centre will undertake field research at sites around the world.
© flickr/Thomas Haw (CC BY-NC-2.0)
The way a society organizes its economy cannot be detached from its ethical concerns and values. In the long period before the emergence of industrial capitalism, accumulation of material goods was seldom the mark of the morally upright. How did the ownership of goods come to be seen as desirable and virtuous? Max Weber asked and answered this question in "The Protestant Ethic and the Spirit of Capitalism" by connecting economic to religious and ethical change, suggesting that major transformations in society are always accompanied by profound changes in what it means to be a good person.
The institutional ramifications of Weber's claims remain hotly debated in the contemporary social sciences. The new Max Planck - Cambridge Centre for Ethics, Economy and Social Change (Max-Cam) will join these debates by applying the ethnographic research methods of socio-cultural anthropology. Drawing inspiration not only from Weber but a range of classical and contemporary social theory, Max-Cam researchers have a particular interest in how ritual practices – from collective celebrations of success and achievement to routinized techniques of accountability and self-cultivation – facilitate the institutionalization of ethical and economic changes.
"In the wake of the global financial crash, when there is great public interest in the ethical foundations of a capitalism dominated by transnational corporations, our goal is to develop new approaches to pressing social agendas," emphasize the four co-Directors James Laidlaw and Joel Robbins (Department of Social Anthropology, University of Cambridge), Chris Hann (Max Planck Institute for Social Anthropology, Halle/Saale), and Peter van der Veer (Max Planck Institute for the Study of Religious and Ethnic Diversity, Göttingen).
Cooperation beyond Brexit
The Max Planck - Cambridge Centre for Ethics, Economy and Social Change is a joint project of these institutions that started its operations in July 2017. "The strengths of the Cambridge Department in the anthropology of ethics, the Göttingen Institute in the study of religion, and the Institute in Halle in economic anthropology, constitute a unique mix of skills with which to investigate contemporary social change," say the Directors.
The Centre is funded jointly by the University of Cambridge and the Max Planck Society and has an initial budget of £2,000,000 (circa 2,270,000 Euro). Six postdoctoral fellows will undertake field research at sites around the world. "We hope in the future to expand our activities , e.g. by offering positions for visiting scholars, and to make the Centre a hub for further creative initiatives beyond the life-span of the initial projects," say the Directors.
Over the coming four years, public lectures, workshops and larger conferences will be hosted both in Cambridge and the two German locations. In the wake of Brexit, the Centre will demonstrate the value of continued European collaboration in science and society.
19 Max Planck Centers worldwide
The Max Planck Centers constitute a major instrument in the Max Planck Society's strategy for international cooperation. The participating Max Planck Institutes and their international partners bundle their resources and typically draw on complementary theoretical traditions and methods to create added scientific value. With the newly inaugurated Max Planck - Cambridge Centre for Ethics, Economy and Social Change there are currently 19 Max Planck Centers worldwide.
JFL/JE
Prof. Dr. Chris Hann
Max Planck Institute for Social Anthropology, Halle (Saale) +49 345 2927-201 hann@eth.mpg.de
Prof. Dr. Peter van der Veer
Max Planck Institute for the Study of Religious and Ethnic Diversity, Göttingen +49 551 4956-226 vdvOffice@mmg.mpg.de
Dr. Johannes Lenhard
Max Cam Centre Coordinator jfl37@cam.ac.uk
Jens Eschert
Press Officer Administrative Headquarters of the Max Planck Society, München +49 89 2108-1488 eschert@gv.mpg
The speakers at the official launch ceremony
Tuesday 6th March, 11:30 am - 1:00 pm, Maudslay Hall of the Museum of Archaeology and Anthropology in Cambridge
Website of Max Planck - Cambridge Centre for Ethics, Economy and Social Change
Max Planck Center und Partnerinstitute
The Max Planck Centers constitute a substantial reinforcement of the international cooperation with first-class international partners. more
Clouds from every angle
Climate Earth Sciences International
The EUREC4A field study is aimed at solving one of the great mysteries in relation to climate change
Social status beats money
Psychology Social Sciences
If a lie could harm their image, people are more likely to forgo financial benefits
Third Call for Dioscuri Centres
The third call for the establishment of up to three centres of scientific excellence in Poland is now open
The Rich, the Poor and Social Cooperation
Rich people are believed to be more cooperative than poor people, an economist from the Max Planck Institute for Tax Law and Public Finance found.
Social inequality in Bronze Age households
Archaeogenetic analyses provide new insights into social inequality 4000 years ago: nuclear families lived together with foreign women and individuals from lower social classes in the same household.
Highest mortality risks for poor
Ageing Social Sciences
Large dataset shows that income, work status and education have a clear influence on mortality in Germany
Eve's Achilles Heel
For women, stress and competition is a bad combination
Research collaboration in Europe
Ceremonial opening of the first two Dioscuri Centres in Warsaw
Genetic diversity of South America
Genetics Language Social Sciences
New study explores genetic roots of 26 populations from diverse regions and cultures of western South America and Mexico, revealing long-distance connections between speakers of the same language, and new traces of genetic diversity within Amazonia.
Support for returning researchers in Argentina
International Research Policy
The Alexander von Humboldt Foundation, the Max Planck Society and the National Scientific and Technical Research Council in Argentina (CONICET) are bundling their support for Argentinian junior researchers returning to their home country
Joining the Square Kilometre Array
Astronomy International Research Policy
Max Planck Society becomes newest member of SKA Organization
Multigene test predicts depression risk
Risk score based on many genetic differences predicts which children and adolescents are at greater risk of depression
"Emphasize social and joint culture"
Lisa Suckert of the Max Planck Institute for the Study of Societies argues for a more differentiated perception of the Brexit campaign and the economic identity of Great Britain.
Second Call for Dioscuri Centres
The second call for the establishment of up to four centres of scientific excellence in Poland is now open (Closed: 20.03.2019)
Dying in hospital
Declining trend, but not for everyone
Influential authors
76 Max Planck scientists among the most highly cited researchers 2018
Success: patience pays off
Matthias Sutter from the Max Planck Institute for Research on Collective Goods in Bonn researches what constitutes success in scientific terms.
The heads of the first Dioscuri Centres
Aleksandra Pekowska and Grzegorz Sumara will each set up a group in Warsaw
Closing the gender gap in competitiveness
The degree of willingness among men and women to assert themselves in competition can be balanced out.


text:
No easy tradeoffs
What does each side "get" in a breakup? What's the real price of immigration? Table Talkers sound off this week.
Salon Staff
October 20, 2006 2:11PM (UTC)
Families Who Think
Divorce/Separation w/ and w/o Children, Part 5
Gibson Girl - 08:53 am Pacific Time - Oct 18, 2006 - #3051 of 3055
Let me tell you, no matter what he "gets," you get your life back.
My ex got his new bike, new house, new pool, all new furniture, redecorated in ways that I had been begging to do for years but was denied because he "hated" it. He kept all of our friends, and his new house has turned into the place to go for parties. It's as if he is some kind of golden boy who never has to take responsibilities for the mess he left behind him, and it used to chap my ass.
I got to keep my 2 best friends, the little old shitty house, old car, old furniture, all the animals (and expenses therein).
However. Big, giant however ... I have my house. It's in my name only, in a good little area with good neighbors, and I'm close to the lake and can walk to get everything I could possibly need (even a sushi restaurant!) I am slowly remodeling, fixing things he refused to deal with, making improvements, doing things the way I want to do them. I used to hate the house. I really like it now, and am planning to stay here for a very long time.
I have my dignity. I stood up to an a--hole and told him "no more, get out." And I got myself out of a bad situation. My self-respect is in full force, I'm free of his soul-crushing presence and I am as independent and confident as I've ever been. After a year to myself, I found a great guy who treats me with kindness, respect, love, humor and affection. We just moved in together.
He "gets" his new house and new car and new life, but the thing he can't "get" is away from himself. He is still the miserable bastard that treated me so badly. Only now he doesn't have someone to blame for his problems, so he'll be forced to deal with them himself.
He may put on a good show about being so wealthy and happy and buying all this new stuff, but I know that it's not true. He's a miserably lonely loser who spends all of his time at home with his cat, smoking weed and watching bad movies on his giant new HD plasma TV and home theater system.
Meanwhile, I have a guy who really cares about me. I've made a whole crowd of new friendly acquaintances, taken up sailing, gone back to my painting and creative pursuits, gone back to school (and I'm 9 hours away from my master's degree) and gotten a LIFE.
"We are America": Views on immigration
Vera Charles - 08:16 am Pacific Time - Oct 13, 2006 - #7 of 50
There are two issues here. One is xenophobia: not liking another culture coming into your own.
The other issue, which I think nearly all liberals ignore almost always, is that there are millions of people in this country who work hours and conditions OSHA would not allow for wages no one else will (or should) work for. There are millions of people whose children are not fluent in English who do not pay income tax to support the schools that now have to provide ESL programs. In my city, the people that hurt are not white people. They are not middle-class people -- middle-class people employ illegal immigrants. They are other dark-skinned and disenfranchised people who themselves need intensive early education and who cannot afford to be displaced by people with stronger families who work tougher jobs for lower wages.
As a private citizen, I teach illegal immigrants all the time, and I love those kids and their families and their music and their tasty food. Good for me. That's not the point. The point is, as a fiscal liberal, I think we need a welfare state, and I don't think we're going to get it if we start giving those benefits to people who are not Americans. My life would be better in Canada. I have an expensive health condition, I make very little money, my kids will be zoned to a terrible school. Am I allowed to go to Canada and live in the kind of socialist democracy I'd like to? Not likely. Is that a racist decision on the part of the Canadian government?
Look, these are good people coming here. Great people. People whose kids do deserve a better life. But our school went 90% Latino once it went 40% because the African-American parents feel run out. We were founded to help inner-city kids, and now we help intact families on their way up the class ladder. It's a great situation for the kids and a great situation for the teachers because, frankly, these kids are a hell of a lot easier to teach, but is it right? Is it right that affirmative action policies are now benefiting students who are not, in this part of the world, a minority, and who are displacing the descendants of slaves?
MORE FROM Salon Staff
Check out this article! https://www.salon.com2006/10/20/post_163/

redpajama_set_name: RedPajamaCommonCrawl


text:
Fernando Santillana Figueroa (died 1574) was a Roman Catholic prelate who served as Bishop-Elect of La Plata o Charcas (1572–1574).

Biography
On 17 October 1572, Fernando Santillana Figueroa was appointed during the papacy of Pope Gregory XIII as Bishop of La Plata o Charcas.
He died before his consecration as Bishop-Elect of La Plata o Charcas in 1574.

References

External links and additional sources
 (for Chronology of Bishops)
 (for Chronology of Bishops)

16th-century Roman Catholic bishops in Bolivia
Bishops appointed by Pope Gregory XIII
1574 deaths
Roman Catholic bishops of Sucre

redpajama_set_name: RedPajamaWikipedia


text:
Комбина́тор неподви́жной то́чки (или оператор неподвижной точки) — функция высшего порядка, вычисляющая неподвижную точку другой функции.

Наиболее известным комбинатором неподвижной точки является Y-комбинатор в λ-исчислении, введённый известным американским учёным Хаскеллом Карри как

Иногда имя этого комбинатора ошибочно используется для обозначения вообще всех комбинаторов неподвижной точки.

Языки программирования, в которых допустим комбинатор неподвижной точки, позволяют использовать рекурсию анонимных функций без присвоения значения такой функции переменной.

Теорема о неподвижной точке
И в λ-исчислении, и в комбинаторной логике для каждого терма  существует по крайней мере один терм  такой, что . И более того, существует комбинатор  такой, что

См. также
 Комбинаторная логика
 Лямбда-исчисление
 Функциональное программирование

Литература
 Вольфенгаген В. Э. Комбинаторная логика в программировании. Вычисления с объектами в примерах и задачах. — М.: МИФИ, 1994. — 204 с.; 2-е изд., М.: АО «Центр ЮрИнфоР», 2003. — 336 с. ISBN 5-89158-101-9.
 Mayer Goldberg, (2005) On the Recursive Enumerability of Fixed-Point Combinators, BRICS Report RS-05-1, University of Aarhus
 Matthias Felleisen. A Lecture on the Why of Y .

Лямбда-исчисление
Комбинаторная логика


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text:
HomeSportsBasketballMen's BasketballRediscovering Chris Herren: The Star BC Lost And The Kid It Forgot
Rediscovering Chris Herren: The Star BC Lost And The Kid It Forgot
September 24, 2014 Connor Mellas Men's Basketball
The Former BC Basketball Player Talks About His Confusing, Difficult Time On Campus
May 1, 1995: When Chris Herren came to BC, everyone expected great things. But after only one game in a BC uniform, his career here is apparently over … A lot has been speculated, and much has been written about the situation. But the bottom line is that Chris Herren missed an opportunity, and he will be fortunate to get another.
It's the first day of fall, Sept. 23, 2014. A hint of autumnal crispness marks the air, but the lingering humidity of late summer is securely trapped inside the Reggie Lewis Track and Athletic Center in Roxbury, M.A. In front of a set of bleachers filled with 300 football-jersey sporting, junior-high kids from some of the roughest neighborhoods in Boston, Chris Herren is in his element.
Turning to his left and right, Herren makes eye contact with every corner of his audience. Then, after a long pause, he asks, "What is it about me on Friday and Saturday nights that I have to change myself?"
Over the last four years, Herren has told his story and posed a question like this to more than 500,000 kids. The middle schoolers in front of him now are part of Boston's Play Ball! Foundation, and just like every other group of kids Herren's stood in front of, he does his best to save them from a nightmare by sharing his own.
His tale is one of throat-welling desperation and eventual, incredible salvation, and despite his tragic link to BC, most students have never even heard of Herren.
Nov. 15, 1993: Boston College basketball arrived on the national scene Wednesday, on what has been called coach Jim O'Brien's most important day here at BC. With the signing of The Boston Globe's Division 1 player of the year, Chris Herren, O'Brien and his staff have put together one of the top five early-signing recruiting groups in the nation.
About three years ago, ESPN Films chronicled Herren's story in the 30 for 30 Unguarded. Equal parts horrifying and inspiring, Unguarded details Herren's journey from star high school basketball player, to prized recruit, to drug-addicted pro, to heroin junkie, to the motivational speaker he is today.
Nov. 14, 1994: The one exception in the group may be 1994 McDonald's All-American point guard Chris Herren out of Durfee High in Fall River, MA, who may just end up being the Eagles' point guard for the next four years.
He was featured on a Sports Illustrated spread and thrown out of BC. Herren starred at Fresno State, and checked into rehab during his junior season. He married his childhood sweetheart and became a father. The Nuggets drafted him 33rd overall in the 1999 NBA Draft before trading him to the Boston Celtics, at which point he became addicted to Oxycontin. After getting injured, the Celtics cut him and he embarked on a career as an international journeyman, falling deep into the darkness of a heroin addiction. Switching to needles over time, he shot up for six years, overdosing and dying for 30 seconds on June 4, 2008. Herren collected seven felonies. He spent every single dollar he made from basketball on drugs, and pawned anything he could find in his house. He overdosed four times in total. And yet, somehow, Herren climbed out of the abyss. On August 1, 2008, he got sober.
Heights File Photo
For one of the most incredible stories to ever reside—albeit very briefly—on Lower Campus, Herren's anonymity among BC's collective consciousness is utterly astonishing. If Herren's story ended with his terrible spiral into drug and alcohol addiction, forgetting him would be less remarkable—that tale's been told a thousand times. But it didn't end there, and a defining part of it began at BC.
Back in high school, Herren drank alcohol, popped some medicine-cabinet painkillers with his friends, and smoked marijuana. Shortly after beginning his career in Chestnut Hill, Herren tried cocaine for the first time in a BC dorm room. He sat down at a table, snorted a line with a dollar bill, and became an addict. Instantly, Herren was flying down a whole new, entirely sinister path, with a paralyzing inability to slow his descent. His problems snowballed quickly. In his first and only game as an Eagle in Conte Forum, Herren excelled—putting up 14 points in 21 minutes. Then he fell, breaking his wrist. He knew he was in trouble. Three failed drug tests later, Herren was done at BC.
Dec. 5, 1994: Boston College men's basketball was dealt a severe blow last week when it was announced that freshman guard Chris Herren will miss the rest of the season. Herren, the heralded guard from, Fall River, MA, will undergo surgery tomorrow to repair ligament damage to his left wrist, according to an announcement by BC head coach Jim O'Brien.
Taking a seat and laughing, Herren shakes my hand and jokingly asks if this interview doubles as a peace offering with BC. It is. At a fit 6-foot-2, 195 lbs, Herren is a bit bigger than he was as an 18-year-old freshman—but given the pharmaceutical hell his body was subjected to, the resemblance to the old photos is striking. Wearing jeans, a white v-neck, and a blue and black blazer, Herren leans back in his chair. I tell him I want to talk about his time at BC, and his face becomes more solemn—he's told these stories countless times before, but the memories remain painful.
The Heights: Would you mind talking about your BC time at all? I know you've probably talked about it a lot, obviously—but if you wouldn't mind just kind of telling me what that was like, going back to then.
Herren: Confusing. I wasn't ready at 18 years old to accept the responsibility of being a big-time college basketball player and being a college student at the same time. It was extremely hard being local and rooted in this area and staying focused at the same time. I had a lot of outside influence around me back then.
The Heights: Out of curiosity, where did you live on campus?
Herren: I was supposed to live in a freshman dorm on Upper Campus, and I walked into my dorm room and [my roommate], who was a Canadian scholarship athlete, had life-sized pictures of The Crow on the wall. So I checked out and got a single across from the new cafeteria.
The Heights: On Lower?
Herren: On Lower Campus. But I spent most of my nights in Edmond's on the first floor, corner window.
The Heights: I've seen Unguarded and I thought it was extremely inspirational—so I guess that's when things really started to get out of control. What was your relationship like with Dick Kelley at that time?
Herren: Dick Kelley was way more than a SID. He was concerned about my wellbeing more so than my ability. We had many, many meetings in his room, in his office, where we discussed the issues. As I said in articles—he was one of the first people I called when I celebrated a year sober. Because he was one of the first people that understood how hard my addiction was. He was right there on the front lines with me in the beginning.
The Heights: Other than breaking your wrist, what do you remember from that first game, that single game?
Herren: You know, it was a blur to me. People told me the stats, but I don't remember scoring, I don't remember any of that stuff. Any time you come from Durfee High School, playing in front of that many people, you're Massachusetts Player Of The Year three years, you're a McDonald's All American, and now it's time to put up or shut up on the big stage—the emotions were through the roof. I remember the injury. I remember falling that night, and knowing that this was probably going to be the end of it.
The Heights: You could tell with that much certainty, at that point?
Herren: As far as basketball. Addiction is like a very personal journey, lonely journey. And I knew young, and I just told NPR, that I was on a self-destruct path when I was at Boston College. And it had nothing to do with Boston College, and it had everything to do with me. I often feel bad that it happened at Boston College because BC gave me such an amazing opportunity to not only be an athlete locally—a free education, a great education—and I blew it. And I take full responsibility for that. It would have happened at Harvard, it would have happened at MIT, and it would have happened at UCLA. I would have opened that dorm room door somewhere, because that's who I was in a sense, that's the path I was on.
Herren spoke at an event for Boston's Play Ball! Foundation. Emily Fahey / Heights Editor
The Heights: After the first failed test, what happened on BC's side, how did they reach out for you—did they reach out for you? And then after the other tests as well?
Herren: You know honestly, I think back then it was pretty uncommon. It wasn't as much of an issue as it is today, so when it happened, a lot of people didn't know how to really handle it, as well as myself and the staff at BC. The trainers tried tremendously to help me. I just wasn't ready, I mean it was not gonna happen. I don't remember the exact way they took care of it. We're talking 20 years ago, and a lot happened in them 20 years, my man, you know what I mean? My memory's shot, my back's shot, brother. But, you know, I think we all could have handled it better. I know I could have.
Do I think BC could have handled it differently? Absolutely. Do I think if it was 2014—I think it would have been addressed a little differently. Drug addiction in athletics—and it still is to a certain extent—is treated very punitively. We punish, we punish, we punish. We suspend, we suspend, we suspend—rather than, why not treat? Why not treat? We send these athletes away to these centers, out of their element, for 30 days, and then we throw them right back on to a college campus and say, "stay sober." Like, did you stay sober at Boston College?
The Heights: Nope.
Herren: Right. You know what I mean? It's the hardest place to ask somebody—anybody—to go spend four years of your life on a college campus and don't drink alcohol. And don't do drugs. I mean, it's difficult. Especially for someone like me, it was a very difficult thing to try to do.
The Heights: And I guess especially when basketball's out of the equation—I'm sure you had a lot of free time, obviously.
Herren: I broke my wrist, which was a nightmare. But I didn't handle it right. I'd walked into BC with a chip on my shoulder, with an air of cockiness that helped me on the court but was not going to help me in the culture. And just living on Edmond's as a freshman, living in Edmond's as a freshman—big mistake. I remember playing Wiffle Ball in the Mods, and people walking by saying, 'Dude, don't you have class?' And I'm like, 'No, I'm playing Wiffle Ball.' But, like, the Mods are seniors getting ready to graduate. And I'm like starting off my college career learning how to throw to hit the corner of the chair in the Mods. You know what I mean? That's a bad start. But I love the Mods.
The Heights: I think everybody does.
Herren: Exactly.
Editor's Note: This interview has been condensed and edited for clarity.
Featured Image by Emily Fahey / Heights Editor
About Connor Mellas 85 Articles
Connor Mellas is a senior at Boston College. He used to be Sports Editor. Now he wanders aimlessly through the void. Follow him on Twitter @MellasHeights.
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This page refers to the comic named "2018". For comic #2018, see 2018: Wall Art.
Title text: We should really start calculating it earlier, but until the end of December we're always too busy trying to figure out which day Christmas will fall on.
The last panel expresses a misunderstanding of modern public-key cryptography, which relies on the fact that it is difficult to factorize large numbers. Megan is applying this concept to the year, claiming that it is hard to determine whether or not 2,018 is a multiple of four and hence is a leap year. In reality, factorization is not needed here, since we already know the factor in question, which is four. Megan states that, if it were possible to factor large numbers with a calculator, modern cryptography would collapse. While true, it is true only for truly large numbers (hundreds of digits), and no factorization is needed in this case.
At the end of the strip, Megan hopes the answer can be brute-forced by February. Brute force is a method of breaking cryptography by trying every possible option until one works. This is misdirection upon misdirection, in that, even if we needed to factorize 2,018 (which we don't), the simplest brute-forcing algorithm would need to try only 14 numbers -- each prime from 2 to 43 (the square root of 2,018 is closest to 44). In cryptography, the algorithms use numbers much, much bigger than 2,018 -- on the order of hundreds or even thousands of digits.
The title text refers to calculating which day Christmas will fall on. As Christmas always lands on December 25 by definition, the day of the week varies from year to year, though it's always the 359th or, in leap years, the 360th day of the year. Still, determining which day of the week December 25 lands on is not a difficult problem to solve, requiring only a few mathematical operations to compute. Alternatively, this might be an oblique reference to Easter, the date of which jumps from year to year according to a multi-layered algorithm that most people don't know. The changing date of Easter was recently included in 1930: Calendar Facts.
Megan: I wonder if 2018 will be a leap year.
Cueball: ...it won't be, right?
Megan: I doubt anyone knows at this point.
Cueball: No, it's definitely not. Leap years are divisible by 4.
Megan: Right, and for odd numbers, that's easy.
Megan: But 2018 is even.
Cueball: I can settle this with a calculator.
Megan: No way. If it were easy to factor large numbers like that, modern cryptography would collapse.
Megan: I just hope we manage to brute-force it by February.
Released on Friday, December 29, this was the last comic of 2017. The next comic, 1936: Desert Golfing, was released soon after midnight (in Randall's time zone) on New Year's Day 2018.
Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400. For example, the years 1700, 1800, and 1900 were not leap years, but the years 1600 and 2000 were.
Since 100 is divisible by 4, only the last two digits of a number are needed to determine if that number is divisible by 4. So to determine if 2018 is divisible by 4, we only need to check whether 18 is divisible by 4, which is easy.
2018 is not divisible by 4, so the year is not a leap year. 2016 and 2020 are leap years. This is true for both the Gregorian and the Julian calendar. A year is roughly 365.2422 days long.
Eastern Christian Churches celebrate Christmas also on December 25 but on the older Julian calendar, which currently corresponds to January 7 on the Gregorian calendar.
This is the third year in a row with New Year's comics with only the year used as the title; before that there were two more comics with such titles, but those two (and thus the first three) were only released in the even years: 998: 2012 in 2012, 1311: 2014 in 2014, 1624: 2016 in 2016 and 1779: 2017 in 2017.
Cueball: No, it's definitely not. Leap years are divisible by 4. Megan: Right, and for odd numbers, that's easy. Megan: But 2018 is even.
This page was last edited on 8 August 2018, at 21:43.


text:
\section{Introduction}
\label{sec:intro}

P/2016~G1~was discovered by the Pan-STARRS1 (PS1) telescope on 2016 Apr.~1, and the orbit ($a$ = 2.582 au, $e$ = 0.210, $i$ = 10.969$^{\circ}$, $T_{\rm Jup}$ = 3.367), suggested that this was possibly a main belt comet \citep{weryk2016}.  The object was approaching a perihelion of $q$ = 2.041 au on 2017 Jan. 26.24. Because the surrounding dust coma had a peculiar appearance in the PS1 images, we undertook an immediate campaign to follow this up with images from the 3.6m Canada-France-Hawaii Telescope (CFHT). The first CFHT images, obtained on 2016 Apr. 3 showed a structure to the south, 90 degrees from the dust tail.  Because of the unusual appearance, we dedicated time from our CFHT PS1 discovery follow up program to image P/2016~G1\ frequently during the dark runs when the wide field imager was on the telescope.  By mid-April, its diffuse and core-less appearance made it clear that the object was undergoing some sort of catastrophic disruption.

\citet{hsieh18} reported that P/2016~G1\ is linked to the Adeona family, which likely originated in a cratering event $\sim 700$~Myr ago \citep{benavidez12, carruba16, milani17}. The Adeona family members are typically C- and Ch-type, while the surrounding background objects are predominantly of the S-type \citep{hsieh18}.

\section{Observations \& Data Reduction}
\label{sed:observations}

\subsection{Pan-STARRS1}
\label{sec:PS1}

With a well-known orbit and a brightness estimated to be above the PS1 limiting magnitude over most of its orbit, we conducted a search through the PS1 database for pre-discovery images. On 2016 Mar. 7 the object is clearly visible, apparently mostly stellar (see below for a more detailed assessment).
{Photometry was obtained using PS1's intrinsic calibration.}
A set of four $w$-band (\B{central wavelength} $\lambda = 6250$~\AA, width 4416~\AA) images on 2016 Jan. 10 shows nothing at all, with a limiting magnitude in the stack of $r\sim$23.
This corresponds to a nucleus radius of 0.81 km for an albedo of 4\% or 0.32 km for an albedo of 25\%.

The position of P/2016~G1\ was imaged 12 additional times in the PS1 data prior to discovery, between 2011 Feb. 23 and 2015 Jan. 17 and nothing was visible down to the limiting magnitudes shown in Table~\ref{tab-data}. The most constraining observations (2012 Jun.) suggest that the object must have a radius $R_N<0.4$~km for a 4\% albedo or $R_N<0.2$~km for 25\% albedo. For these non-detections, a positional uncertainty region of 3-$\sigma$ was inspected.  These limits are thus valid under the assumption that the existing astrometric arc is good and that a gravity-only solution models the orbit well enough over 5 years.

\begin{figure}[ht]
\includegraphics[width=8cm]{Fig1_orbit_P2016G1.pdf}
\caption{
Distribution of observations (see Table 1) of P/2016~G1\ along its orbit. The symbols indicate the apparitions, from aphelion (Q) to aphelion. Perihelion is marked (q). Filled symbols show when the comet was detected, open symbols where it was not detected. \B{No observations were acquired during the 2002 Jul. -- 2006 Sep. apparition}. The positions of the planet symbols correspond to 2016 Jul.
}
\label{fig:orbit}
\end{figure}

\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=17.5cm]{Fig-P2016G1a.pdf}
\includegraphics[width=17.5cm]{Fig-P2016G1b.pdf}
\includegraphics[width=17.5cm]{Fig-P2016G1c.pdf}
\caption{\small
[A] Images of P/2016 G1 between UT 2016 March 7 and July 4.  All the images are scaled to be 6$\times$10$^4$ km on a side. The Earth crossed the object's orbital plane on Apr. 13, so images around that date show the coma extension above and below the plane.
[B] Images of P/2016 G1 that have been processed with unsharp masking to enhance
\B{the three clumps and the details} near the disintegrating body.  Each image is 1.5$\times$10$^4$ km on a side.
[C] Contour plots on three dates showing the expansion of the southern arc and the eastern linear feature (highlighted with red lines). Each panel is $30''\times 20''$. The panel widths and pixel scales are: April 3 (34,000~km, 227~km), April 14 (32,000~km, 231~km) and May 12 (29,000~km, 193~km). North is up, East is Left. The Sun and velocity vectors are listed in Table~1.
}
\label{fig:images}
\end{center}
\end{figure*}

\subsection{Canada-France-Hawaii Telescope}
Imaging data were obtained using the MegaCam wide-field imager which is an array of 40 2048$\times$4612 pixel CCDs, with a plate scale of 0$\farcs$187 pixel$^{-1}$.  The epochs and circumstances of the CFHT data are listed in Table~\ref{tab-data}. MegaCam is on the telescope for a period centred on each new moon, and data are obtained through queue service observing and are processed to remove the instrumental signature through the Elixir pipeline\footnote{http://www.cfht.hawaii.edu/Instruments/Elixir}.

The photometric calibration of the processed data accesses the Pan-STARRS database or SDSS to provide a photometric zero point to each frame, using published colour corrections \citep{tonry12} {to translate PS1 $g,r,i,z$ bands into SDSS or Johnson-Cousins} systems.
The object headers are used to identify the target and download orbital elements from the Minor Planet Center; the resulting object location is used to determine which object in the frame corresponds to the target.  In the final pass, Terapix tools \citep[SExtractor,][]{SEx} are run to produce multi-aperture and automatic aperture target photometry.

A first series of CFHT observations was obtained as an immediate follow-up to the discovery, showing the dust features that constitute the core of the following discussion. A second was acquired with the broad $gri$ filter on 2018 Dec. 12 and 31, over 1000~days after the first series. The object has completely dispersed, showing no visible remain down to $gri=26$ ($5\sigma$ point-source, using the MegaCam Direct Imaging Exposure Time Calculator\footnote{http://cfht.hawaii.edu/Instruments/Imaging/  MegaPrime/dietmegacam.html}.

\subsection{Himalayan Chandra Telescope (HCT)}

Observations of P/2016 G1 were made on 2016 May 31 using the Himalayan Chandra 2.0-m telescope with the Himalaya Faint Object Spectrograph and Camera (HFOSC). The instrument uses a SITe ST-002 CCD with a plate scale of 0$\farcs$296 per pixel.  Data were obtained through Bessell $R$-band filters under clear conditions with seeing of $\sim$1$\farcs$6.

We processed the data using our image reduction pipeline which bias-subtracts and flattens the data, and then applies the Terapix tools to fit a precise WCS to the frame. Following the removal of instrumental effects, the photometric calibration proceeded in the same manner as for the CFHT data.

\subsection{Observatory Archives and other publications}
\label{sec:archives}

Additional pre-discovery images were found from the CFHT (2007 Feb. 16) and the INT on La Palma (2000 Oct. 20), both with non-detections, both less constraining than the 2012 Jun. images. For these older images, the positional uncertainties were $\pm$45$''$ and $\pm$3$'$, respectively.

Finally, the Canadian Astronomy Data Centre's Solar System Object Image Search \citep{soss} was used, and about 40 short exposures with the Cerro Tololo Interamerican Observatory 4-meter telescope were identified. Based on the 2016 Jan. 10 upper limit, and accounting for the geometry of the orbit, these CTIO images would fail to reach the same limit by 1--2~mag.

The positions on the orbit for the dates for which we have observations (including both detections and non-detections) are shown in Fig.~\ref{fig:orbit}. This shows that we have tightly bracketed the period of apparent activity for this current apparition, strongly suggesting that whatever caused the activity began between 2016 Jan. 10 and 2016 Apr. 1 when P/2016 G1 was discovered, and that the object did not present significant activity during previous apparitions.

\cite{moreno2016} have acquired images with the 10.4~m Gran Telescopio Canarias on 2016 Apr. 21, May 29 and Jun. 08. The general appearance of their images fits well with our CFHT images discussed below. They followed up with two observations obtained with the Hubble Space Telescope on 2016 Jun. 28 and Jul. 11 \citep{moreno2017}, which focus on the central region of the disrupted object. From these deep, high-resolution images, they estimate that no fragment larger than $\sim 30$~m survived. They modelled the observations using their Monte Carlo dust-tail code; their results are discussed in the comparison with ours in Section~\ref{sec:moreno}.

\begin{figure}[ht!]
\includegraphics[width=8.5cm]{Fig-SB.pdf}
\caption{\small
Average surface brightness profile of P/2016 G1 compared to the average profile for field stars as seen in the data obtained with the Pan-STARRS1 telescope on 2016 March~7.  The profiles for the field stars have been normalised to the peak brightness for P/2016 G1.  Although the images of the comet appear stellar in the \B{images, its profiles indicate that the object is slightly --but significantly-- extended.}
}
\label{fig-phot}
\end{figure}

\section{Analysis \& Results}
\label{sec:results}

\subsection{Surface Brightness Profiles}
\label{sec:SB}

The pre-\B{dis}covery data on P/2016~G1\ with the PS1 telescope on 2016 Mar. 7 consist of four images, showing the object with a stellar morphology. Only two of the frames had the object clear of the chip gap boundary. The surface brightness profile was computed in each of these frames for P/2016~G1\ and for three nearby field stars in order to assess whether there was any dust surrounding the nucleus.  We compute the radial flux profile using the median per-pixel flux in circular annuli around the object, and convert the flux to surface brightness using the zeropoint in the PS1 headers.  Errors are obtained using bootstrap re-sampling within each annulus. Figure~\ref{fig-phot} presents the average normalised surface brightness profiles of both P/2016~G1\ and the field stars, and shows that P/2016~G1\ is more extended. During each exposure, the object moved about 0$\farcs$3, which is significantly less than the seeing, so that trailing does not cause the flux excess seen in P/2016~G1. We conclude that the object was already surrounded by dust on 2016 Mar.~7.

\subsection{Description and nomenclature}\label{nomenclature}

Figure \ref{fig:images} shows a series of images of the object over time.
The central structure of the object is composed of 3 main concentrations of dust forming an inverted ``C'' on the images \B{(hereafter referred to as the three clumps)}, and a much fainter concentration on the eastern side, leaving a central gap. This structure is growing with time, preserving its appearance. The central structure is bathed in a diffuse, elongated coma, that extends into a dust tail westward from the central structure. The orientation of the coma and the tail is rotating southward as the viewing geometry evolves.

Below the central structure, a small, narrow, well-defined arc extends toward the South-South-East. It is clearly seen from Apr.~3 until May~12 and is still detected on May~31. Directly eastward of the central structure, but not pointing toward its centre, a faint linear feature is present. It can also be seen until May~12. The arc and the linear features are marked on Fig.~\ref{fig:images}C.

\subsection{Main Head Dust Feature\label{head}}
\label{sec:head}

\begin{figure}[ht!]
\includegraphics[width=8.5cm]{G1_size.pdf}
\caption{\small
Size of the main head structure as a function of time. The line and the horizontal error bar indicate the linear expansion fitted to the data and the error on the origin, 2016 Mar. 6 $\pm$ 3~days.
}
\label{fig:size}
\end{figure}

The Earth crossed the comet's orbital plane on 2016 Apr. 13. The images taken around that date indicate that the complex head structure and the central coma extends above and below the orbital plane, whose orientation corresponds on the images to that of the tail feature (PA$=274^\circ$). The grains forming the head structure must, therefore, have been ejected with a non-zero velocity, with a non-zero component perpendicular to the orbital plane.

The size of the central structure was estimated measuring the distance between the peak of the North and South concentrations. When the peak is not well defined, the centre of the concentration was used instead. These angular distances were converted to a linear distance in the plane of sky accounting for the geocentric distance (and reported in Fig.~\ref{fig:size}). In the Mar.~7 image, \B{although the object is broader than stellar,}the central structure is not resolved yet. The diameter of the seeing disk is therefore used as an upper limit. The size of the structure grew linearly with time. A linear regression indicates that the growth would have started on 2016 Mar.~6 (JD=2457454) $\pm$ 3 days, and is expanding with an average (plane of sky) velocity of 0.32$\pm$0.02 m s$^{-1}$ (i.e. drifting from the centre of the original object at half that speed). The Mar.~7 image would therefore have been obtained very soon after the onset of the activity, \B{the observed broadening of the image could be caused by the cloud of fine dust expanding more rapidly than the larger fragments.}

\subsection{Finson-Probstein Dust Modelling\label{FPmodel}}

The positions of the Earth, Comet and Sun in late May and in June show the dust features of the tail with a geometry very favourable for a Finson-Probstein \citep[FP][]{finson68, farnham96} dust dynamical analysis.  FP modelling calculates the trajectories of an ensemble of dust grains of different sizes, parameterised by $\beta$ (ratio between the radiation pressure and the solar gravity) ejected from the surface of the nucleus at different times, $\tau$, as acted upon by solar gravity and solar radiation pressure.  We used the FP approach to analyse the pattern of synchrones (loci of the particles emitted at the same time) and syndynes (curves joining particles with the same $\beta$) and the optical appearance of the comet's dust environment.

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=8.5cm]{G1_FigFP_20160531.pdf}
\includegraphics[width=8.5cm]{G1_FigFP_20160608.pdf}
\end{center}
\caption{\small Syndynes (in green) and synchrones (in purple, labelled in days before the observation date) for P/2016 G1 on 2016 May 31 and 2016 Jun. 8. The thicker synchrone marks the brightest peak in the dust profile, and the corresponding grain radii are marked in red. This fits with the disruption taking place around March 6.
}
\label{fig:fp}
\end{figure}

The main tail-like feature has a fairly sharp profile in azimuth as shown in Fig.~\ref{fig:fp}.  The position angle of the peak was measured on the images; it is marked as a thick line on the FP plots.  The epoch of the corresponding synchrone was obtained by comparing the position angles of synchrones generated with a step of 1 day.  The error on that epoch resulting from the measurement uncertainty is $\pm$3 days.  The azimuthal profile of the tail is very roughly gaussian; converting the FWHM (measured using the same procedure as above), this also results in a ``$\sigma$'' of $\pm$3 days.  The broadening of the tail is caused in part by the seeing (negligible far from the nucleus), by the dispersion in emission velocity (which is neglected in the zero-velocity FP method) and by the duration of the dust emission. The broadening caused by the emission velocity can be estimated from the images obtained when the Earth was close to the orbital plane (when the synchrones and syndynes degenerate into a single line), around Apr.~13 (see Fig.~\ref{fig:images}): \B{the width of the tail on Apr.~14 is similar to that seen on Apr.~4--8, when the orbital plane was observed at an angle.} We conclude that the broadening of the tail is dominated by the velocity dispersion of the grains \B{(which occurs in all directions, including perpendicular to the orbital plane), rather than purely by the radiation pressure (which occurs only within the orbital plane)}.

The tail-like feature is therefore compatible with a burst of dust emission centred on 2016 Mar.~4$\pm$3 days for the image from 2016 May 31, and 2016 Mar.~7$\pm$3 days for the image from 2016 Jun. 8. The emission profile is compatible with a short, impulsive burst smeared out by a distribution of initial velocities of the particles with $v < 1$\ms\ in random directions, or a longer burst (increasing then decreasing over a few days) with no initial velocity. Furthermore, dust is present in the areas of the image not covered by the syndynes and synchrones, confirming that the dust must have been emitted with some initial velocity. We therefore favour the short, impulsive burst.

It is interesting that two independent measurements for the date of the onset of disintegration: the FP estimates and the central feature growth are in perfect agreement.

Assuming that the tail-like structure is dominated by dust emitted with zero velocity, the spread of the dust along the tail is caused only by the radiation pressure. Close to the head of the object, the actual velocity dispersion of the grains will smear this relation, but further away, $\beta$ (the ratio between the radiation pressure and the solar gravitation for the considered grain), $a$ (the grain radius) and $\rho$ (its density) are related by:
\begin{equation}
\beta = 5.740 \times 10^{-4} \times \frac{ Q_{\rm pr}}{ \rho a} .
\end{equation}
\noindent
$Q_{\rm pr}$ is a radiation pressure efficiency coefficient, in the 1--2 range for rocky and icy material. With this relation, we can estimate the characteristic grain sizes in the images: with $\rho = 3000$~kg~m$^{-3}$, and $Q = 1.05$, Eq. 1 gives $ a = 2\times 10^{-7} / \beta$~[m].  The loci of 10 to 200~$\mu$m grains are marked on Fig.~\ref{fig:fp}. Closer to the head than the 100$\mu$m mark, the blurring by the seeing and the non-zero velocity of the grains forming the head structure prevent FP from making meaningful estimates of the grain size beyond ``they are large'', {\it i.e.} in the mm range or even larger.

\subsection{Dynamical Model of the Arc and Linear Feature\label{HeadModel}}

\begin{figure}
\includegraphics[width=8.5cm]{bubbleSky.pdf}
\caption{\small
The positions of large particles emitted by the nucleus on 2016~Mar.~6, with velocities ranging from 0 to 3\ms\ observed on 2017~May~12 are marked with light grey dots in the plane of sky. The coordinates mark offsets from the position of the nucleus for that time. For clarity, only a subset of the particles is plotted. The particles matching the position of the 3 head features are marked in blue. Those matching the arc are marked in red (the ends of the feature are marked in magenta and black); the blob at the western start of the linear feature (Fig.~\ref{fig:images}) is marked in orange.
\label{fig:bubbleSky}
}
\end{figure}
\begin{figure*}
\includegraphics[width=18.8cm]{bubbleMap.pdf}
\caption{\small
For a series of emission velocity, maps of the emission direction (in cometocentric longitudes and latitudes) of the large particles matching the features observed on May 12, using the same colour code as in Fig.~\ref{fig:bubbleSky}. For any velocity, a particle matching a feature always has a "mirror particle": one is emitted toward the Earth, the other one away from the Earth. On the 2.375, 2.5 and 2.625\ms\ panels, the position of a $40^\circ$ half-opening cone matching best the feature is represented. The direction directly toward Earth is marked by a black cross, away from Earth by a red cross, at $0^\circ$ latitude, and $90^\circ$ and $270^\circ$ longitude respectively. Re-projecting these maps onto concentric spheres, with the geometry of May~12, would lead to Fig.~\ref{fig:bubbleSky}.
}
\label{fig:bubbleMap}
\end{figure*}
\begin{figure}\begin{center}
\includegraphics[width=7.0cm]{G1_2016Apr03.pdf}
\includegraphics[width=7.0cm]{G1_2016May12.pdf}
\includegraphics[width=7.0cm]{G1_2016May31.pdf}
\end{center}
\vspace{-0.2cm}
\caption{\small
P/2016 G1 on 2016~Apr.~3, May~12 and 31. The green dots mark clumps on the arc feature. The small red dots correspond to the synchrone trajectories of small dust (submitted to radiation pressure) emitted from some of the clumps on Mar.~6. The position of large particles emitted on Mar.~6 on a cone with half-opening 40$^{\circ}$ and centred on (long., lat.) = (68$\deg$,-15$\deg$) are over-plotted for velocities ranging from 1.625 to 2.750\ms. The solid line corresponds to 2.5\ms.
\label{fig:coneImages}
}
\end{figure}

In order to further investigate the structure of the head, we simulated the trajectory of large particles (i.e. not influenced by radiation pressure, $\beta = 0$), emitted from the nucleus' position on 2016~Mar.~6 with velocity values ranging from 0 to 3~m~s$^{-1}$, and directions covering the whole sphere every 2.5$^{\circ}$ in longitude and latitude \B{(cometocentric coordinates, $x$ and $y$ in the orbital plane), $y$ pointing toward the Earth)}. The position of each particle was computed for 2016~Apr.~3, May~12, and May 31. The particles that were located at the position of the 3~fragment clumps, the  South-East arc and the start of the linear feature are identified, as illustrated in Fig.~\ref{fig:bubbleSky}. Figure~\ref{fig:bubbleMap} shows the maps of the particles matching the features (in cometocentric longitude and latitude at the time of the emission), for a set of emission velocities.

{\bf The three main fragment clumps} can be reproduced by particles emitted at any velocity between 0.25 and 3\ms. However, for higher velocities, they must be emitted almost exactly toward or away from Earth (so that the foreshortening keeps them at their position near the origin), which is unlikely. It is more likely that they were emitted at a random angle, i.e. at velocities $<$ 1\ms.

{\bf The South-East Arc} cannot have been emitted at velocities $<$ 2.25\ms: part of the arc is not reproduced (some of the dots are missing). For velocities in the 2.25 -- 3\ms\ range, the complete feature can be reproduced by particles emitted at the same velocity. A cone with a half-opening of $40^\circ$, whose axis points toward cometocentric coordinates (long., lat.) = (68$\deg$,-15$\deg$), is found to reproduce well the arc feature and the start of the linear feature, with a velocity of 2.5\ms: the arc and the clump at the head of the linear features are on a ring corresponding to the 2.5\ms slice of this same cone.

In order to test the validity of this interpretation, the position of the particles was also computed for Apr. 3 and May 31, when the viewing geometry was very different. As illustrated in Fig.~\ref{fig:coneImages}, the same ring of dust matches well the 3D position and evolution of the arc and the head of the linear feature.

Adjusting the position and opening of the cone (by eye) gives an estimate of the uncertainty on these parameters: $\pm 2.5^\circ$ in longitude and latitude, and $\pm 5^\circ$ in opening. Changing the opening of the cone also slightly modifies the velocities of the particles matching the features. The date of the initial burst was also changed by 5, 10, 15~days. No effect is noticeable for 5~day; a slight change of orientation and velocity is required for the 10 and 15~day shifts. This therefore does not set additional constraints on the time of the emission.

Using the FP formalism (Section~\ref{FPmodel}), the trajectory of small dust grains emitted together with the larger grains was calculated accounting for the radiation pressure. The smaller grains form a line (synchrone) starting at the large particle, and drifting away at an angle set by the emission date (while the large particles move on a straight line in the image plane, from the nucleus toward their observed position, the small particles start tangentially to that line, and drift on a parabolic particle toward their observed position).  \B{The eastern linear feature matches well the position of small grains emitted at the same time, in the same direction and at the same velocity as the clump at the head of the linear feature. They are highlighted with red dots in the May~12 panel of Fig.~\ref{fig:coneImages}.}
Similarly, various clumps in the arc feature have similar ``tails'', which together cause an enhancement of surface brightness on the western side of the arc \B{(two of them are highlighted with red dots in the May~12 panel of Fig.~\ref{fig:coneImages})}. As the position angle of these synchrone lines is set only by the time of the emission, the excellent match between the eastern linear feature and the synchrone for a Mar.~4 emission gives a very strong support to the hypothesis that it corresponds to small grains emitted simultaneously and in the same direction as the big grains, and further support the Mar.~4 emission date. It also rules out that the linear feature would have been caused by a secondary disruption of a large grain at a later date: the position angle of the corresponding synchrone would have been different.

Overall, the southern arc and the head of the eastern linear feature are large particles partially populating a ring. The arc covers at least a quarter of the ring, possibly more, as its Northern part is lost in the main coma. A single clump of large particles on the ring forms the tip of the eastern linear feature. This ring grows in size with time, as the particles move away from the emission spot at $(2.5 \pm 0.1)$\ms, describing a cone with an half-opening of 40$^{\circ}$.

It is possible that the sections of the ring that appear devoid of dust were originally populated by small grains: solar radiation pressure would have pushed them away from the field of view.

The ring corresponds to a very narrow velocity distribution: in particular, the ``walls'' of the cone are not populated by dust grains, only a (partial) ring.

We tried and failed to reproduce the arc and the eastern linear feature using dust grains emitted along a great circle in cometocentric coordinates, as would have been produced by an equatorial centrifugal ejection. We also tried and failed to reproduce the observed features using grains with a broader range of velocities: a velocity distribution around 2.5~\ms, narrower than 0.125~\ms, gives the best results.

\subsection{The Moreno Model}\label{sec:moreno}

\citet{moreno2016} obtained data for P/2016 G1 on three nights between 2016 Apr.~20 and Jun.~8 and used Monte Carlo techniques to model the dust. They obtained two additional epochs with the Hubble Space Telescope on 2016 Jun.~28 and Jul.~11 \citep{moreno2017}.

Comparing the original dataset to model images generated using Monte-Carlo emitted particles following a FP-like formalism, they infer that the dust ejection began around 2016 February 10$^{+10}_{-30}$, and decreased with a half-Gaussian function with a half-width at half maximum (HWHM) of $24^{+10}_{-7}$~d, corresponding to an emission of 1.7$\times$10$^7$ kg of dust, of which some was preferentially ejected in the westward direction, consistent with an impact aligned with the sun-asteroid vector. They inferred very low dust velocities with average speeds around \B{0.8\ms}\ and grain sizes between 1$\mu$m to 1~cm in radius\B{(the velocities in the original paper \citet{moreno2016} had a typo, corrected in an erratum \citet{moreno2019})}. The more recent HST data are compatible with their original model.

Their analysis was focused on what we call here the central structure with the 3 main concentrations forming the inverted C on the images and the central dust coma. From an FP analysis similar to that of Section~\ref{FPmodel}, they set the interval for the beginning of activity in their model from 2016 Jan.~31 to Apr.~1.

Our estimate of the date of the peak of dust emission relies on two simple and independent measurements:
$i)$ the linear growth of the separation between the condensations in the central structure, which agrees with the barely resolved image obtained on Mar.~7 (Section~\ref{sec:SB}), and
$ii)$ the position angle of the sharp cusp in the isophotes far from the origin, in Fig.~\ref{fig:fp}. Measuring this angle on Moreno's Fig.~2, we get a very good agreement with our estimate. This makes the simple assumption that far from the origin, the effect of the radiation pressure dominates over an initial velocity. Moreno et al. run their Monte-Carlo multi-parameter model to refine the combination of initial velocity and radiation pressure, leading to a difference in emission time of about a month.

Their analysis does not mention the southern arc, nor the eastern linear features, whose geometry we find to strongly and independently confirm the emission date provided by the orientation of the main tail and the growth of the central structure, and to also support a very brief emission. This is further confirmed by the very narrow eastern linear feature, that points precisely at an impulsive emission on again the same date, Mar.6. Because of the convergence of these independent arguments, because they are based on much wider features than those Moreno et al. worked on, and because of the simplicity of the underlying assumptions, we are confident that the disruption occurred on 2016 Mar. 6$\pm$3~days.

In terms of duration, the analysis of Moreno et al. suggests that ``[an] impact would have induced a partial destruction of the asteroid,'' causing the anisotropic part of the dust emission, ``with dust grains being emitted to space nearly isotropically while the body is being torn apart,'' that this isotropic emission would have had a duration of $24^{+10}_{-7}$~d HWHM. Our data directly support the impact-induced destruction of the asteroid,  the arc and linear feature being the signature of that impact. Concerning the broad dust feature on which Moreno et al. focus their study, our interpretation is that the broadening of the tail matches the dust velocity dispersion, and that therefore it is compatible with a very short duration event, although we cannot rule out (nor constrain) a longer duration for the final crumbling of the asteroid.

\subsection{Mass of the debris cloud}

\begin{figure}
\includegraphics[width=8.7cm]{G1_mass.pdf}
\caption{\small Top: radius of the dust grains as a function of the distance to the centre of the object, using the FP scaling (see Sect.~\ref{FPmodel} and Fig.~\ref{fig:fp}) for May 31. Bottom: the flux in the May 31 image is integrated over concentric rings and converted into the number of particles (blue), and in mass (red)\B{, both in each radial bin}.
}
\label{fig:mass}
\end{figure}

Because the exact grain size distribution present in each pixel of the image is not known, and because the central structure can contain very large particles (meter-sized or more), getting an evaluation of the mass of the object is difficult. This is made worse by the fact that larger particles contribute more to the mass, while smaller particles contribute more to the flux. Below comes an attempt to get a rough estimate of this mass, based on the May~31 image. We assume a dust density $\rho = 3000$~kg~m$^{-3}$, and an albedo $p=0.25$.

To get an order-of-magnitude estimate, one can consider a uniform size distribution across the whole object (a power law with a $-3$ index, and a cut-off at 1~cm, after \citet{moreno2016}). This leads to a total mass in the object of $1\times 10^7$~kg, in agreement with Moreno et al. who obtained an ejected mass of $2\times 10^7$~kg with their detailed model.

As we consider that the dust release event was short, the dust distribution (far from the position of the nucleus) is dominated by the radiation pressure: smaller grains are pushed further away than larger ones. While this estimate is also a simplification, it takes into account the fact that the dust is not uniformly distributed. The grain size is set using the scaling between size and distance from Fig.~\ref{fig:fp}. The image is then integrated on concentric rings, and the flux is converted in dust surface, then in the number of grain, then in mass, as illustrated in Fig.~\ref{fig:mass}. This results in a total mass for the object $M_N = 1.1\times 10^9$~kg. This estimate is still a lower limit, as larger clumps could hide in the central concentration. The mass corresponds to a spherical object with a radius $R_N \sim 44$~m, well below the limit of detection of the pre-discovery images ($R_N<200$ m).

We don't know whether the object was {\em completely} disrupted. Using their high-resolution, deep HST images, \citet{moreno2017} obtained an upper limit of 30~m to the fragments. The mass potentially contained in even a few of these large fragments would completely dominate the mass of the dust. In what follows we consider as a lower limit the mass of the dust (and the corresponding radius $R_N > 44$~m), and the upper limit from the pre-discovery constraints ($R_N < 200$~m). Most of the mass being located in the central concentration, which we observed expanding at $v=0.16$\ms. The corresponding kinetic energy is $K=1.4\times 10^7$ and $1.3 \times 10^9$~J, respectively. In order to reach this asymptotic velocity, the grains and fragments had to overcome the (small) gravity potential of the object, with an escape velocity $v_e = \sqrt{\frac{2GM}{r_T}} = 0.06$\ms\ and $0.26$\ms, respectively. This implies that the disruption of the object required a total energy at least $E = \frac{1}{2} M_N (v^2 + v_e^2) = 1.6 \times 10^7$ and $4.7 \times 10^9$~J, respectively. These are clearly lower limits that consider a pure rubble pile, i.e. without and internal cohesion forces.

Considering that this energy was provided by an impactor, and that collision happened at the median impact velocity in the main belt $v_i= 5\,000$\ms, this requires a tiny impactor of 1~kg, with a radius of 5~cm for the lower limit on the object size, and 370~kg with a radius of 30~cm for the larger object.

\subsection{Summary of the analysis}\label{sec:analysissummary}

Overall, our results are based on various independent measurements and separate simple interpretation pointing toward the following scenario:
\begin{itemize}
\item No image from before 2016 Mar. shows the object. These non-detections lead to a radius $R_N< 0.2$~km.
\item The onset of activity was on 2016 Mar. 6. Backward extrapolation of the central structure gives an uncertainty of $\pm$3 days but the object was found active on Mar.~7. This Mar. 6 date is confirmed by the following independent measurements: orientation of the azimuthal peak of the main tail; position and evolution of the southern arc and of the head of the linear feature; orientation of the eastern linear feature.
\item The dust emission event had a very brief component (indicated by the narrow eastern linear feature and the narrow ring of large particles populating the southern arc), which can possibly have been followed by a longer emission of dust from the central feature.
\item The central component of the object is composed of large dust grains moving at low velocity ($0.16\pm0.01$\ms\ in the plane of sky) from the original centre. It is surrounded by a cloud of smaller particles, also emitted at similar low speed, and being pushed away by radiation pressure. The visible tail is populated by particles in the 10--200~$\mu$m range. Larger particles are certainly present, but confused into the central component; smaller particles drifted away from the field of view. \B{The broadening of the Mar.~7 image could be caused by the hyper-velocity material released during the impact itself, while the slower material was released during the follow-up disruption}.
\item During the brief emission event, a group of particles was emitted on a
cone with a half-opening angle of $40\pm5^\circ$, propagating at (2.5$\pm$0.1)\ms and forming part of a ring, with an arc covering over a quarter of the ring plus a clump opposite to the arc (forming the southern arc and the head of the eastern linear feature). The large particles were emitted together (in time, direction and velocity) with smaller particles, which are seen drifting from the ring under the radiation pressure. They form the eastern linear feature and a faint westward extension to the southern arc. These features could not have been reproduced by emission on a grand circle (matching, for instance, an equatorial ejection).
\end{itemize}

\section{Discussion: Impact \& Disruption\label{discussion}}

From the observations, a major disruption occurred on P/2016~G1\ on or around 2016 Mar. 6, disrupting --possibly destroying-- the object, and releasing a ring of large particles moving at $\sim 2.5$\ms, growing on a cone with a half-opening angle of $\sim 40^\circ$. We consider the impact of a meter-scale object hitting a 100-meter-scale asteroid at $\sim 5$~\kms. All the hyper-velocity ejecta which would be expanding at impact-scale speeds (in the \kms\ range) have moved degrees away from the position of the object at the time of the first deep images (about one month after the impact). Radiation pressure having further dispersed these grains, they are lost beyond the field-of-view of the observations.

All that is left in the field-of-view are the spalls and large dis-aggregated fragments of the objects that came off well after the impact, in the final stages of the formation of the crater and after, together with the dust grains released.
\B{Crater formation simulations (numerical and laboratory) show that asymmetries in the filling of the ejection cone are common; this would explain why the ring is not complete.}

These grains ($>10\mu$m) are either the small end of a power-law size distribution of the fragments, and/or the regolith that was dragged along with bigger fragments. The bulk of the impactor's energy is transferred into the bulk of the body, causing its disruption into fragments that drift apart at very low velocities.

Hydrodynamic simulations of impacts can provide information on the remnants, in particular on their size distribution \citep[see][for a systematic study of various parameters]{durda2007} and velocities. Simulations of catastrophic impacts\footnote{
URLs: https://youtu.be/pqaqbfbevoo \\
https://youtu.be/e4HCTcQ-IWA} show that the plate-shaped spall fragments, corresponding to the final moments of the crater formation, are ejected with different velocities and over a broad range of directions. However, the crater formation process also produces wedge-shaped fragments, bounded by the propagating radial cracks. These have a narrow distribution of slow velocity, in the \ms\ range, and are axially isotropic about the point of impact. These large wedges would become visible as they break apart and their surface area increase. Based on these qualitative arguments, we suggest that the feature we interpret as a ring of fragments corresponds to the left-over of these large wedge-shaped fragments. Hydrodynamics simulations, beyond the scope of this paper, will be used to quantitatively verify the plausibility of this hypothesis. Interestingly, the velocity of ejection of these wedge-shaped fragments is directly linked to the tensile strength of the object. The tensile strength is one of the key structural characteristics of an asteroid, but it is normally not accessible remotely. In the case of a rotationally disrupted asteroid, the study of the fragmentation can cast some light on the internal strength, but with some uncertainties caused by the unknown initial spin of the object \citep[see][for an application to rotationally disrupted P/2013~R3]{hirabayashi14}.
A result of the ongoing simulation will, therefore, be the first direct remote estimate of the tensile strength of an asteroid, a measurement that would have implication for the mitigation measures being investigated for planetary defence initiatives.

\begin{acknowledgements}
We are very grateful for the detailed review and comments provided by the referee, Fernando Moreno. KJM, JK, and JVK acknowledge support through an award from the National Science Foundation AST1413736.  RJW acknowledges support by NASA under grant NNX14AM74G. This paper is based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. Data were acquired using the PS1 System operated by the PS1 Science Consortium (PS1SC) and its member institutions. We thank the staff of IAO, Hanle and CREST, Hosakote, that made these observations possible. IAO and CREST are operated by the Indian Institute for Astrophysics, Bangalore. The Pan-STARRS1 Surveys (PS1) have been made possible by contributions from PS1SC member Institutions and NASA through Grant NNX08AR22G, the NSF under Grant No. AST-123886, the Univ. of MD, and Eotvos Lorand Univ. We would like to thank Detlef Koschny (ESA) and Mikael Granvik for their helpful discussion on impact cratering mechanics and disruption. This research was supported by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence ``Origin and Structure of the Universe''.
\end{acknowledgements}

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