Asymptotics in Astrophysics Study Group

The aim of this study group is to bring together physicists and mathematicians to share techniques and tricks from the textbook "Advanced Mathematical Methods for Scientists and Engineers" by Bender and Orszag, which are used to solve complex physics problems. This group aims to facilitate the exchange of valuable mathematical maneuvers across different fields, particularly those involving wave-related physics.

Objectives

Basic idea of study group

The “asymptotics and perturbation theory for astrophysics” study group will bring together physicists and mathematicians who work with asymptotics and perturbation theory techniques in their respective fields. Specifically, applications of the techniques presented in the pioneering textbook: Bender, C. M. ; Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill (New York), 1978.

Context

Utilising these methods to solve physics problems generally requires special tricks or transformations, or physically motivated simplifications. These tricks are seldom communicated across fields, but would be invaluable information to learn. This study group is therefore being proposed as a unique opportunity to share these kinds of secret mathematical manoeuvres and manipulations.

For example, over the past 50 years, the groundbreaking textbook: Unno, W. ; Osaki, Y. ; Ando, H. ; Shibahashi, H., Nonradial oscillations of stars, University of Tokyo Press (Tokyo), 1979 was used as the basis of the development of the field of asteroseismology (e.g. The Kavli Prize 2022 in Astrophysics, The Crafoord Prize in Astronomy 2024).

Thanks to asymptotics and perturbation theory, it is now common-use to analyse stellar interiors (convective properties, rotation speeds) experimentally via observed oscillation freqencies (which manifest as periodic brightening). This was a truly remarkable breakthrough, given that stars are completely opaque at their surface.

At iTHEMS, there are several senior researchers who have developed, or are currently developing novel techniques in asymptotics and perturbation theory in e.g. high energy physics, cosmology. Junior researchers in e.g. astrophysics can therefore take the opportunity to learn from them, by first introducing specific applications of these techniques in our respective fields.

In astrophysics, while asymptotics and perturbation theory has been successfully applied to understand stellar oscillations via asteroseismology (see also e.g. the long term stability or instability, chaos of three body systems which interact gravitationally), there are several more recent problems in astrophysics where attempts to utilise asymptotics and perturbation theory are either inconsistent with astrophysical observations in nature, or inconsistent with eachother.

Some of these wave dynamics and/or linear stability related problems include:

How much mass can ubiquitous, convection driven internal gravity (buoyancy) waves eject at the surface of a massive star in the final months before it core-collapses? →And what should we see with currently operating telescopes before, during and after a supernova?
How efficiently can internal gravity (buoyancy) waves redistribute angular momentum within a star?
How efficiently can magnetic field line breaking redistribute angular momentum within a star? →what does this mean for the expected black hole or neutron star (pulsar) spin rates formed by core collapse in massive stars?

Facilitators:: Lucy McNeill (RIKEN iTHEMS) *Contact at lucy.mcneill@riken.jp; Masazumi Honda (RIKEN iTHEMS)