̽��ֱ�� of Cambridge - Faculty of Mathematics

Testing the water

skbf2 — Thu, 30 May 2024 09:31:04 +0000

Industry placements for PhD students can be hugely beneficial for all concerned. We hear from two former students and the companies they worked for.

̽��ֱ��Vice-Chancellor's Awards 2023 for Research Impact and Engagement

zs332 — Wed, 13 Dec 2023 09:20:46 +0000

Meet the winner of the Vice-Chancellor's Awards 2023 for Research Impact and Engagement and learn more about their projects.

Cambridge launches 'Problem-Solving Schools' initiative as AI means key skill is even more crucial

sb726 — Thu, 16 Nov 2023 11:45:00 +0000

Mathematicians at the ̽��ֱ�� of Cambridge are supporting UK schools to help prioritise problem solving in maths.

Astronomers use ‘little hurricanes’ to weigh and date planets around young stars

sc604 — Fri, 06 Jan 2023 08:25:43 +0000

Researchers from the ̽��ֱ�� of Cambridge and the Institute for Advanced Study have developed a technique, which uses observations of these ‘hurricanes’ by the Atacama Large Millimeter/submillimetre Array (ALMA) to place some limits on the mass and age of planets in a young star system.

Pancake-like clouds of gases, dust and ice surrounding young stars – known as protoplanetary discs - are where the process of planet formation begins. Through a process known as core accretion, gravity causes particles in the disc to stick to each other, eventually forming larger solid bodies such as asteroids or planets. As young planets form, they start to carve gaps in the protoplanetary disc, like grooves on a vinyl record.

Even a relatively small planet – as small as one-tenth the mass of Jupiter according to some recent calculations – may be capable of creating such gaps. As these ‘super-Neptune’ planets can orbit their star at a distance greater than Pluto orbits the Sun, traditional methods of exoplanet detection cannot be used.

In addition to the grooves, observations from ALMA have shown other distinct structures in protoplanetary discs, such as banana- or peanut-shaped arcs and clumps. It had been thought that at least some of these structures were also driven by planets.

“Something must be causing these structures to form,” said lead author Professor Roman Rafikov from Cambridge’s Department of Applied Mathematics and Theoretical Physics, and the Institute for Advanced Study in Princeton, New Jersey. “One of the possible mechanisms for producing these structures – and certainly the most intriguing one – is that dust particles that we see as arcs and clumps are concentrated in the centres of fluid vortices: essentially little hurricanes that can be triggered by a particular instability at the edges of the gaps carved in protoplanetary discs by planets.”

Working with his PhD student Nicolas Cimerman, Rafikov used this interpretation to develop a method to constrain a planet’s mass or age if a vortex is observed in a protoplanetary disc. Their results have been accepted for publication in two separate papers in the Monthly Notices of the Royal Astronomical Society.

“It’s extremely difficult to study smaller planets that are far away from their star by directly imaging them: it would be like trying to spot a firefly in front of a lighthouse,” said Rafikov. “We need other, different methods to learn about these planets.”

To develop their method, the two researchers first theoretically calculated the length of time it would take for a vortex to be produced in the disc by a planet. They then used these calculations to constrain the properties of planets in discs with vortices, basically setting lower limits on the planet’s mass or age. They call these techniques ‘vortex weighing’ and ‘vortex dating’ of planets.

When a growing planet becomes massive enough, it starts pushing material from the disc away, creating the tell-tale gap in the disc. When this happens, material on the outside of the gap becomes denser than material on the inside of the gap. As the gap gets deeper and the differences in density become large, an instability can be triggered. This instability perturbs the disc and can eventually produce a vortex.

“Over time, multiple vortices can merge together, evolving into one big structure that looks like the arcs we’ve observed with ALMA,” said Cimerman. Since the vortices need time to form, the researchers say their method is like a clock that can help determine the mass and age of the planet.

“More massive planets produce vortices earlier in their development due to their stronger gravity, so we can use the vortices to place some constraints on the mass of the planet, even if we can’t see the planet directly,” said Rafikov.

Using various data points such as spectra, luminosity and motion, astronomers can determine the approximate age of a star. With this information, the Cambridge researchers calculated the lowest possible mass of a planet that could have been in orbit around the star since the protoplanetary disc formed and was able to produce a vortex that could be seen by ALMA. This helped them put a lower limit on the mass of the planet without observing it directly.

By applying this technique to several known protoplanetary discs with prominent arcs, suggestive of vortices, the researchers found that the putative planets creating these vortices must have masses of at least several tens of Earth masses, in the super-Neptune range.

“In my daily work, I often focus on the technical aspects of performing the simulations,” said Cimerman. “It’s exciting when things come together and we can use our theoretical findings to learn something about real systems.”

“Our constraints can be combined with the limits provided by other methods to improve our understanding of planetary characteristics and planet formation pathways in these systems,” said Rafikov. “By studying planet formation in other star systems, we may learn more about how our own Solar System evolved.”

̽��ֱ��research was supported in part by the Science and Technology Facilities Council (STFC), part of UK Research and Innovation (UKRI).

References:
Roman R Rafikov and Nicolas P Cimerman. ‘Vortex weighing and dating of planets in protoplanetary discs.’ Monthly Notices of the Royal Astronomical Society (2022). DOI: 10.1093/mnras/stac3692 or DOI: 10.48550/arXiv.2301.01789

Nicolas P Cimerman and Roman R Rafikov. ‘Emergence of vortices at the edges of planet-driven gaps in protoplanetary discs.’ Monthly Notices of the Royal Astronomical Society (2022). DOI: 10.1093/mnras/stac3507

Little ‘hurricanes’ that form in the discs of gas and dust around young stars can be used to study certain aspects of planet formation, even for smaller planets which orbit their star at large distances and are out of reach for most telescopes.

It’s extremely difficult to study smaller planets that are far away from their star by directly imaging them: it would be like trying to spot a firefly in front of a lighthouse. We need other, different methods to learn about these planets

Roman Rafikov

ALMA (ESO/NAOJ/NRAO)

ALMA image of the protoplanetary disc around HL Tauri

̽��ֱ��text in this work is licensed under a Creative Commons Attribution 4.0 International License. Images, including our videos, are Copyright © ̽��ֱ�� of Cambridge and licensors/contributors as identified. All rights reserved. We make our image and video content available in a number of ways – as here, on our main website under its Terms and conditions, and on a range of channels including social media that permit your use and sharing of our content under their respective Terms.

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Cambridge Maths School: Chance to find out more as applications open for 2023

sb726 — Wed, 05 Oct 2022 16:28:18 +0000

A Cambridge Maths School Open Event will offer school pupils an opportunity to find out more about the new pioneering sixth form, which opens next year.

Chance to find out more about pioneering Cambridge Maths School at open days

sb726 — Mon, 27 Jun 2022 09:59:46 +0000

̽��ֱ��new specialist, state-funded sixth form - opening in Mill Road, Cambridge, in September 2023 – is being developed in partnership with the ̽��ֱ�� of Cambridge, where the open days will be hosted. ̽��ֱ��free events, on Saturday, 9 July and Tuesday, 12 July are aimed at Year 10 students, but are open to all students who are starting to consider their A-level options.

̽��ֱ��Eastern Learning Alliance (ELA) - a multi-academy trust with schools across Cambridgeshire and East Anglia – will run the Cambridge Mathematics School, where talented students from across the East of England will study maths and further maths, and then choose from physics, chemistry, biology or computer science A-levels.

Working with the ̽��ֱ��’s Faculty of Mathematics, and drawing on its outreach and widening participation expertise - in particular the success of the Millennium Mathematics Project (MMP) and its NRICH programme - the Cambridge Maths School aims to attract more female students into maths subjects, more minority ethnic students, and more students from socially and educationally disadvantaged backgrounds. ̽��ֱ��School is also working with the Cambridge Mathematics project - a collaboration between Cambridge ̽��ֱ�� Press and Assessment, and the Faculties of Education and Mathematics - to create an innovative mathematics curriculum.

Clare Hargraves, Headteacher of the Cambridge Maths School, said: “ ̽��ֱ��open days are the first opportunity for many school students to find out about our plans for the School and its pioneering learning, and we are delighted to be able to share our excitement with them. We want to stimulate students’ passion for mathematics through an extraordinary curriculum that connects ideas and inspires young people – we hope as many as possible can come along to the open days and discover more.”

Book tickets for the Cambridge Maths School open days, taking place 9 and 12 July at the ̽��ֱ�� of Cambridge’s Centre for Mathematical Sciences, Wilberforce Road, Cambridge.

*Ahead of the open days in July, the Cambridge Maths School’s first outreach enrichment events are taking place in June, for Year 9 and 10 students. Activities will be provided by the ̽��ֱ��’s NRICH and Isaac Physics outreach programmes, among others. ̽��ֱ��in-person ‘Eureka Days’, at Cambridge’s Centre for Mathematical Sciences, are designed to inspire curious mathematicians. ̽��ֱ��sessions will feature a paper crane physics challenge, ‘Dragon Quiz’, a session designed around the importance of collaboration, and a talk exploring ‘the hidden maths behind the digital world, from World War II to Wi-Fi’ by mathematician and public speaker Dr James Grime, presenter of the YouTube channel Numberphile.

School students from across the East of England will find out more about the new Cambridge Maths School at a series of open day events in July.

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Founding headteacher appointed to lead new Cambridge Mathematics School

sb726 — Thu, 07 Apr 2022 08:24:21 +0000

Maths education expert Clare Hargraves will lead the state-funded specialist sixth-form college, which is being created by multi-academy trust ̽��ֱ��Eastern Learning Alliance (ELA) in collaboration with the ̽��ֱ��’s Faculty of Mathematics.

Clare has maths leadership experience in secondary schools and sixth form colleges across the East of England, including as a deputy head in a school rated in the top 1% of state schools nationally, and during a sixth-form headship where she was responsible for transforming the outcomes of a previously underperforming college. Most recently, her trust-wide consultancy work has focused on curriculum design, the professional development of staff, and embedding a culture of rigorously high expectations in mathematics education.

She said: “It’s an enormous privilege to be the founding headteacher of the Cambridge Mathematics School, but this is of course a collaboration that involves the expertise, passion and experience of many people – including at the ̽��ֱ�� of Cambridge. This partnership is committed to developing pioneering learning and ensuring truly outstanding and enhanced mathematics provision is available to A-Level students across the region.”

̽��ֱ��Cambridge Mathematics School - which will open in Mill Road, Cambridge, in September 2023 - will teach 16 to 19-year-old A-Level students from across the East of England, and aims to attract more female students into maths subjects, more minority ethnic students, and more students from socially and educationally disadvantaged backgrounds. ̽��ֱ��principal aim of maths schools is to help prepare more of the UK’s most mathematically able students to succeed in maths disciplines at top universities, and address the UK’s skills shortage in science, technology, engineering and maths (STEM) subjects.

As part of the academy trust- ̽��ֱ�� collaboration, the School is working with the ̽��ֱ��’s Faculty of Mathematics, and in particular with the Cambridge Mathematics project (a collaboration between Cambridge ̽��ֱ�� Press and Assessment and the Faculties of Education and Mathematics) to create an innovative mathematics curriculum. It is also drawing on the Faculty of Mathematics’ widening participation and outreach experience, in particular the success of the Millennium Mathematics Project (MMP) and its NRICH programme.

Lynne McClure, Director of the Cambridge Mathematics project, said: “There is a wonderful opportunity here to work with high attaining students and their highly qualified and passionate teachers, and to do something different. ̽��ֱ��Cambridge Mathematics team is looking forward to sharing our research and evidence base in the design of an exciting, engaging and cutting-edge curriculum.”

Professor Colm-cille Caulfield, Head of the Department of Applied Mathematics & Theoretical Physics at the ̽��ֱ�� of Cambridge (DAMTP), said: “Mathematics in Cambridge has a rich past and a vibrant present. In the Faculty we aim both to advance mathematical knowledge through novel and insightful research, and to train the next cohort of mathematicians through innovative and rigorous teaching. Through MMP and NRICH we are committed to the development and support of pre- ̽��ֱ�� mathematical education for all, and looking to the future, we are excited to engage with the Cambridge Mathematics School to enable students from all backgrounds to ‘think like a mathematician’.”

Find out more about the Cambridge Mathematics School

̽��ֱ��development of the new Cambridge Mathematics School, in partnership with the ̽��ֱ�� of Cambridge, continues with the appointment of a founding headteacher.

We are excited to engage with the Cambridge Mathematics School to enable students from all backgrounds to ‘think like a mathematician’.

Professor Colm-cille Caulfield, Head of DAMTP

Clare Hargraves, Cambridge Mathematics School

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NRICH: nurturing next-generation problem solvers

ta385 — Wed, 30 Mar 2022 06:00:00 +0000

NRICH spent the last two years in emergency rescue mode, helping learners in lockdown. Its online resources attracted over a million page views per week. Now celebrating their 25th anniversary, the NRICH team is more determined than ever to nurture our next-generation problem solvers.