A general basis set algorithm for galactic haloes and discs

We present a unified approach to (bi-)orthogonal basis sets for gravitating systems. Central to our discussion is the notion of mutual gravitational energy, which gives rise to a 'self-energy inner product' on mass densities. We consider a first-order differential operator that is self-adjoint with respect to this inner product, and prove a general theorem that gives the conditions under which a (bi-)orthogonal basis set arises by repeated application of this differential operator. We then show that these conditions are fulfilled by all the families of analytical basis sets with infinite extent that have been discovered to date. The new theoretical framework turns out to be closely connected to Fourier-Mellin transforms, and it is a powerful tool for constructing general basis sets. We demonstrate this by deriving a basis set for the isochrone model and demonstrating its numerical reliability by reproducing a known result concerning unstable radial modes.

Automated summary

We propose a unified approach to describing (bi-)orthogonal basis sets for gravitating systems. The concept of mutual gravitational energy is central to our discussion, leading to a 'self-energy inner product' on mass densities. We examine a self-adjoint first-order differential operator with respect to this inner product and prove a general theorem that specifies the conditions for a (bi-)orthogonal basis set to arise through repeated application of this operator. We demonstrate the fulfillment of these conditions by existing families of analytical basis sets with infinite extent. Moreover, our new theoretical framework is closely connected to Fourier-Mellin transforms and serves as a powerful tool for constructing general basis sets. To illustrate this, we derive a basis set for the isochrone model and validate its numerical reliability by reproducing a known result on unstable radial modes.