this proper motion corresponds to a velocity of 10 km s$^{-1}$. Thus, although the astrometry systematics in our SDSS-*Gaia* proper motion catalog are small, at the typical distances of our halo stars we cannot robustly measure rotation signals weaker than 10 km s$^{-1}$. We discuss this point further in Section \[sec:res\].

In the remainder of this work, we use Eqn. \[eqn:sig\] to define the proper motion uncertainties of our halo star samples (see below). Thus, we assume that the proper motion errors are random, independent and normally distributed with variance depending only on the time-baseline between SDSS and *Gaia* measurements. Note that since we are trying to measure the *centroid* of the proper motion distribution (i.e. the net rotation), rather than deconvolve it into components or measure their width, we are not very sensitive to knowing the proper motion errors precisely.

Stellar Halo Stars {#sec:samples}
==================

RR Lyrae {#sec:rrl}
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RR Lyrae (RRL) stars are pulsating horizontal branch stars found abundantly in the stellar halo of our Galaxy. These variable stars have a well-defined Period-Luminosity-Metallicity relation, and their distances can typically be measured with accuracies of less than 10 percent. Furthermore, RRL have bright absolute magnitudes ($M_V \sim 0.6$), so 