Neutron star accretion and the neutrino fireball

The mixing necessary to explain the Fe'' line widths and possibly the observed red shifts of 1987A is explained in terms of large scale, entropy conserving, up and down flows (calculated with a smooth particle 2-D code) taking place between the neutron star and the explosion shock wave due to the gravity and neutrino deposition. Depending upon conditions of entropy and mass flux further accretion takes place in single events, similar to relaxation oscillator, fed by the downward flows of low entropy matter. The shock, in turn, is driven by the upflow of the buoyant high entropy bubbles. Some accretion events will reach a temperature high enough to create a neutrino fireball,'' a region hot enough, 11 Mev, so as to be partially opaque to its own (neutrino) radiation. The continuing neutrino deposition drives the explosion shock until the entropy of matter flowing downwards onto the neutron star is high enough to prevent further accretion. This process should result in a robust supernova explosion

PUP Math World series

This video comes from The Private Universe Project in Mathematics and includes interview with researcher, Regina Kiczek, as well as narrative voice-over, interspersed with footage of students engaged with problem solving. The problem asks: In the World Series, assuming two teams are equally matched, and the first team that wins four games wins the series, what is the probability that the World Series will be won in four, in five, in six, and in seven games? The students spend about an hour working on this task. Ankur, Brian, Jeff and Romina work together to find the possible outcomes of each game as a case. Michael works mostly by himself. Then they use the chalkboard to share their approaches and discuss to convince one another of solutions. After Jeff shares what they found using the strategy of cases, Michael calls their attention to a relationship between the number of outcomes for each game and entries in the diagonal of Pascal’s Triangle. Although Michael recognizes the pattern, the students are not yet sure what it means. However, the discovery will become a catalyst for these students to revisit and extend their ideas in subsequent problem-solving sessions.Transcript is also available.Smithsonian Astrophysical Observatory. (1998). PUP Math World Series [video].Resource vailable in QuickTime and Flash digital video formats

PUP Math Shirts and Pants

This edited video from the Private Universe Project in Mathematics focuses on three children, first in second grade and then in third, exploring two problems, the first that involves making outfits from shirts and pants and the second forming place settings with cups, bowls and plates. There are voice-over explanations that illustrate the children's drawings interspersed throughout the video episode, and interpretive narrative given by researchers Carolyn Maher and Amy Martino. The first problem, Shirts and Pants, was presented to the students, working in small groups, in the spring of second grade. Stephanie, Dana and Michael were grouped together for the first session. In the first session, the girls reported five outfits although their conversation and work suggested considering the six possibiliies. When the same problem was presented in the fall of their 3rd grade, the girls worked as a pair and Michael was partnered with another student. All of the children gave thoughtful solutions to the problem. Later in that school year, the students were presented a second combinations problem called Cups, Bowls and Plates. This part of the video focuses on Dana and Stephanie as they solve the problem and represent their solution for sharing. The final piece of the video is a short excerpt from a follow-up interview with Stephanie, conducted by researcher Alice Alston. Shirts and Pants Problem Statement: Stephen has a white shirt, a blue shirt and a yellow shirt. He has a pair of blue jeans and a pair of white jeans. How many different outfits can he make? Cup, Bowls and Plates Problem Statement: Let's pretend that there's a birthday party in your class today. It's your job to set the places with cups, bowls and plates. The cups and bowls are blueall or yellow. The plates are either blue, yellow or orange. Is it possible for ten children at the party each to have a different combination of cup, bowl, and plate?Transcript is also available.Smithsonian Astrophysical Observatory. (2000). PUP Math Shirts and Pants [video]. Retrieved from http://hdl.rutgers.edu/1782.1/rucore00000001201.Video.000062037Resource vailable in QuickTime and Flash digital video formats

PUP Math Towers

This video comes from The Private Universe Project in Mathematics and includes narrative voice-over and interview with researcher, Carolyn Maher interspersed with footage of students engaged with problem solving and discussion with researchers about their work on building towers with unifix cubes available as a math manipulative. It begins when the students are in 3rd grade where they work with a partner on the problem: How many different towers four blocks tall can you build when selecting from two colors? Students work on the task for about an hour on the first day. The task is challenging for them, and it pushes them to invent strategies and heuristics for problem solving. About five minute into the task, the students discover that they need to check for duplicates as they try to find all the possible combinations. This prompts them to develop ways of organizing the towers they build, as they try to convince themselves, their partners, and the researchers that they have found them all and have no duplicates. The researchers return the next day and ask the students to consider building towers three cubes tall selecting from two colors of cubes. They ask the students whether they think there will be more, the same amount, or fewer towers than when they are four cubes tall. Student responses may be surprising, and they are given opportuntity to explore whether or not their conjectures are correct, as well as discover reasons why. Next, the video shows the students as 4th graders working on the task to build towers five cubes tall selecting from two colors and convince others that they have found all possible combinations. As before, the challenging nature of the task gives students opportunity to spontaneously develop new strategies. Asking students to provide convincing arguments as justification for their solutions pushes them to move beyond trial and error. The researchers conduct interviews with students the following day to find out more about what they were thinking, and about the extent to which students are aware of their own thinking.Transcript is also available.Smithsonian Astrophysical Observatory. (2000). PUP Math Towers [video].Resource vailable in QuickTime and Flash digital video formats

PUP Math Pizza, Clip 1 of 2: Pizza halves with two toppings

In this edited video, the first of a set of two clips developed for the Private Universe Project in Mathematics, 12 fifth grade students during two consecutive classroom sessions work in two groups, one with five students and the other with seven, to construct convincing solutions for the Pizza Halves with Two Toppings problem. The students construct various representations, including drawings and symbolic charts to find and justify their conclusions about the number of possible pizza choices. There are voice-over explanations that refer to the children's representations and strategies interspersed throughout the edited video episode, and interpretive narrative by researcher Carolyn Maher and several of the students who reflect about their earlier problem-solving activity several years later as high school students. Problem statement: Capri Pizza has asked us to help design a form to keep track of certain pizza sales. Their standard “plain” pizza contains cheese. On this cheese pizza, one or two toppings can be added to either half of the plain pie or whole pie. How many choices do customers have if they can choose from two different toppings (sausage and pepperoni) that can be placed on either a whole cheese pizza or half of a cheese pizza? List all possibilities. Show your plan for determining these choices. Convince us that you have accounted for all possibilities and that there could be no more.Transcript is also available.Smithsonian Astrophysical Observatory. (1993). PUP Math Pizza, Clip 1 of 2: Pizza halves with two toppingsResource vailable in QuickTime and Flash digital video formats

PUP Math Romina's proof to Ankur's challenge

In this edited and narrated episode from the Private Universe Project in Mathematics, five tenth-grade students consider two different problems. FIRST PROBLEM STATEMENT: “Choosing from two colors of Unifix® cubes, red and yellow, how many total combinations exist for towers 5 tall, that each contains two red? Convince us that you have found them all.” SECOND PROBLEM STATEMENT (Ankur’s Challenge): “How many towers can you build four tall, selecting from cubes available in three different colors of Unifix® cubes, so that the resulting towers contain at least one of each color?” During the session Ankur and Michael work together at one end of the table and Jeff, Romina, and Brian work at the other end. Ankur and Michael use binary notation to solve the first problem. While waiting for Jeff, Romina, and Brian to complete the first tower problem of five-tall with a choice of two colors, Ankur poses a new question about towers four tall with a selection of three colors: “How many with at least one of each color?” Romina develops a notation that she calls “ones, zeroes, and Xs” to represent the three colors in the problem. First at the table and then at the chalkboard, Romina presents her reasoning to the others as she works to convince them that there thirty-six total towers that meet Ankur’s criteria. She shows them her representation to justify the six possible arrangements for two “1s” in the four positions of the tower. She fills the blank positions with “X0” or “0X” for the other two colors. She then explains that the resulting twelve towers would be multipied by three to account for the three different colors of cubes available.Transcript is also available.Smithsonian Astrophysical Observatory. (1998). PUP Math Romina's proof to Ankur's challenge [video].Resource vailable in QuickTime and Flash digital video formats