The Relation Between Transverse and Radial Velocity Distributions for Observations of an Isotropic Velocity Field

TL;DR

This paper analyzes the probability distribution of transverse velocities in an isotropic velocity field, providing formulas to relate it to radial velocities and identifying conditions for maximizing expected transverse velocity.

Contribution

It derives a mathematical relationship between radial and transverse velocity distributions and determines how to choose radial velocities to maximize expected transverse velocity.

Findings

Probability distribution of transverse velocity given radial velocity is derived.

Conditions for maximizing expected transverse velocity are identified.

Formulas relate observable radial velocities to unobservable transverse velocities.

Abstract

We examine the case of a random isotropic velocity field, in which one of the velocity components (the "radial" component, with magnitude $v_z$) can be measured easily, while measurement of the velocity perpendicular to this component (the "transverse" component, with magnitude $v_T$) is more difficult and requires long-time monitoring. Particularly important examples are the motion of galaxies at cosmological distances and the interpretation of Gaia data on the proper motion of stars in globular clusters and dwarf galaxies. We address two questions: what is the probability distribution of $v_T$ for a given $v_z$, and for what choice of $v_z$ is the expected value of $v_T$ maximized? We show that, for a given $v_z$, the probability that $v_T$ exceeds some value $v_0$ is $p(v_T \ge v_0 | v_z) = {p_z(\sqrt{v_0^2 + v_z^2}})/{p_z(v_z)}$, where $p_z(v_z)$ is the probability distribution of $v_z$. The expected value of $v_T$ is maximized by choosing $v_z$ as large as possible whenever $\ln p_z(\sqrt{v_z})$ has a positive second derivative, and by taking $v_z$ as small as possible when this second derivative is negative.