http://arxiv.org/abs/1808.01208
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. 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.
R. Scherrer and A. Loeb
Mon, 6 Aug 18
23/33
Comments: 5 pages, 4 figures
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