A method for reconstructing the variance of a 3D physical field from 2D observations: Application to turbulence in the ISM

TL;DR

This paper presents a new method to estimate the three-dimensional variance of a physical field from two-dimensional observations, tested on turbulence simulations in molecular clouds, with implications for star formation models.

Contribution

It introduces a general, isotropy-based approach to reconstruct 3D variance from 2D data and validates it using turbulence simulations, enabling observational testing of theoretical predictions.

Findings

The method recovers 3D density variance with ~10% accuracy under isotropy.

Isotropy assumption fails at low sonic Mach numbers in sub-Alfvenic turbulence.

3D density variance scales with the square of the Mach number.

Abstract

We introduce and test an expression for calculating the variance of a physical field in three dimensions using only information contained in the two-dimensional projection of the field. The method is general but assumes statistical isotropy. To test the method we apply it to numerical simulations of hydrodynamic and magnetohydrodynamic turbulence in molecular clouds, and demonstrate that it can recover the 3D normalised density variance with ~10% accuracy if the assumption of isotropy is valid. We show that the assumption of isotropy breaks down at low sonic Mach number if the turbulence is sub-Alfvenic. Theoretical predictions suggest that the 3D density variance should increase proportionally to the square of the Mach number of the turbulence. Application of our method will allow this prediction to be tested observationally and therefore constrain a large body of analytic models of star formation that rely on it.