text:
\section{Introduction}

The pattern speed of the central bar of the Milky Way is a fundamental parameter for characterising our Galaxy. As the orbits that support a bar do not exist much beyond corotation, the length of a bar is set by its pattern speed~\citep{Ag98}. We naturally wish to understand how our Galaxy is structured and how it compares to other barred spirals~\citep{BlandHawthornGerhard2016}. The pattern speed is an essential parameter that allows us to make such direct comparisons~\citep{Aguerri2015,Guo2019}. Interpretation of other Galactic observations require a robust understanding of the bar and its resonances -- for example, observations of the solar neighbourhood velocity substructure~\citep{Ka91,Dehnen1999,Mo17} or interpretation of the high velocity peaks seen in radial velocity surveys towards the Galactic Centre~\citep{Mo15,AU15} or analysis of the bimodal distribution of red clump magnitudes~\citep{Na10,McW10}. The pattern speed of the bar is related to our understanding of when and how it formed, and how it has subsequently interacted with other components in the Galaxy. For instance, bars may be slowed significantly via dynamical friction through interaction with a dark matter halo~\citep{De00}, whilst buckling instabilities can transform rapidly rotating bars into more sedately rotating peanut-shaped bars~\citep{Ra91}.

The pattern speed of the bar has proved to be an awkward parameter to pin down. In part, this is because it is not trivially related to the velocity of its constituent stars, but instead describes the rate of figure rotation of the density and potential of the bar. Stars or gas within the bar possess net streaming in the bar frame, as the orbits that support a bar are preferentially prograde rotating \citep[e.g.,][]{BinneyTremaine2008}. \cite{Gerhard2011} summarises the methods used for the measurement of the pattern speed in the Milky Way. Indirect methods include use of hydrodynamical simulations to reproduce features in the Galactic longitude versus CO and HI terminal velocity ($\ell,v$) maps~\citep{Fu99,Bi03}. This relies on the fact that gas traces the closed orbital structure and so is an excellent probes of the gravitational potential of the bar. Another indirect method identifies the Hercules stellar streams in the local stellar velocity distribution with a family of resonant orbits
in a barred potential.
Evidence for the interpretation of the Hercules stream as a resonance has mounted in recent years~\citep{My18,Hu18}, though its causation remains unclear.
There is a dichotomy between models producing the Hercules stream as the outer Lindblad resonance of a short-fast bar \citep{Dehnen1999,An14} and those with a long-slow bar that produce the Hercules stream as a corotation resonance \citep{Pe17} or a 4:1 outer Lindblad resonance \citep{Hunt2018}.
However, the most direct method was introduced by \citet{TremaineWeinberg} and is derived from the continuity equation with minimal assumptions. It has now been successfully applied to many external galaxies~\citep{Me95,Ge99,De02,Aguerri2015}. \cite{Debattista2002} derived a version of the \citet{TremaineWeinberg} method applicable to line of sight velocity datasets in the Galaxy and demonstrated its use on a sample of $\sim 700$ OH/IR stars in the bar.

\cite{BlandHawthornGerhard2016} report a number of measurements of the pattern speed. Recent hydrodynamical modelling has obtained
$\Omega_{\rm p} = 42$ $\mathrm{km\,s}^{-1}$ kpc$^{-1}$ \citep{We99}, 30-40 $\,\mathrm{km\,s}^{-1}$ kpc$^{-1}$ \citep{Co08} and 40 $\mathrm{km\,s}^{-1}$ kpc$^{-1}$ \citep{So15}. Using the Hercules stream tends to give higher values; \citet{Dehnen2000} using {\it Hipparcos} data originally found
$\Omega_{\rm p} = (51\pm 4)$ $\mathrm{km\,s}^{-1}$ kpc$^{-1}$, while more recent studies have obtained $(51.5 \pm 1.5)$ \citep{Mi07} and
$(53 \pm 0.5)$ km\,s$^{-1}$ kpc$^{-1}$ \citep{An14}. The only previous attempt to use the \citet{TremaineWeinberg} method for the Galactic bar gave one of the highest values of the pattern speed of all, namely $\Omega_{\rm p} = (59 \pm 15)\,\mathrm{km\,s}^{-1}$ kpc$^{-1}$ \citep{Debattista2002}.

Additionally, a lowish pattern speed is supported by the fully dynamical models of the stellar populations of the bar. For example, \citet{Portail2015} used Made-to-Measure methods to reproduce the three-dimensional density of red clump giants, as well as the BRAVA line of sight velocity data~\citep[e.g.,][]{Ku12} in selected fields. This concluded that a still lower pattern speed of $\Omega_{\rm p} = 25-30\,\mathrm{km\,s}^{-1}$ kpc$^{-1}$ is favoured. Subsequent hydrodynamical simulations using the same gravitational force field seem to confirm the low pattern speed as providing the best fit to date to the observed ($\ell,v$) data~\citep{Li16}.
Since then, by incorporating additional kinematic data from the OGLE~\citep{Rattenbury2007} and ARGOS surveys~\citep{Ne16} into the Made-to-Measure modelling, \cite{Portail2017} measured a somewhat higher pattern speed of $\Omega_{\rm p} = (39\pm 3.5)\,\mathrm{km\,s}^{-1}$kpc$^{-1}$, consistent with the recommended combined estimate of \cite{BlandHawthornGerhard2016} of $\Omega_\mathrm{p} = (43 \pm 9)$ $\mathrm{km\,s}^{-1}$ kpc$^{-1}$ using  pre-2016 literature estimates.

It is apparent that, despite a lot of effort, the pattern speed of the bar has resisted an easy concensus. Given the advent of new proper motion catalogues from the Gaia satellite~\citep{Gaia2016,Gaia2018} and elsewhere~\citep{Sm18}, now
seems a propitious moment to generalise the \citet{TremaineWeinberg} method to transverse motions and apply it to the Galactic bar anew.

In a companion paper (Sanders, Smith, Evans \& Lucas 2019, hereafter Paper I), we extracted the velocity field of the bar/bulge from the proper motions of $\sim45$ million stars across the Vista Variables in the Via Lactea \citep[VVV,][]{Minniti2010,Saito2012} survey. We used proper motions from a combination of Gaia DR2 and the VIRAC catalogue. The VIRAC catalogue used the multi-epoch data from VVV to compute relative proper motions which were fixed to an absolute frame using Gaia DR2. In this paper, we use the results of Paper I to measure the pattern speed of the bar/bulge. Our data does not cover the long bar which may be rotating differentially with respect to the bar/bulge. In Section~\ref{Section::PatternSpeed} we derive new expressions for estimating the pattern speed from proper motion data using the continuity equation. In Section~\ref{Section::Application} we apply these expressions to the bar/bulge data. In Section~\ref{Section::Recovery} we demonstrate how well the method works when applied to a simulation of a dynamically-formed bar/bulge and test the limitations and assumptions of the presented method. In Appendix~\ref{Appendix:z0} we generalise the expressions for the pattern speed to the account for the Sun's small offset from the Galactic plane, and in Appendix~\ref{appendix:lf} we validate our choice of luminosity function using local Gaia data.

\section{The pattern speed of the bar}\label{Section::PatternSpeed}
\cite{TremaineWeinberg} introduced a method for measuring the pattern speed of a barred disc galaxy using only the continuity equation and the assumption that the pattern is steady.  \cite{KuijkenTremaine1991} and \cite{Debattista2002} adapted the formalism for use with line-of-sight velocities in the Milky Way with \cite{Debattista2002} applying the formulae to OH/IR stars across the Galactic disc. Whilst \cite{TremaineWeinberg} and \cite{KuijkenTremaine1991} worked in 2D, we shall follow \cite{Debattista2002} who provided expressions for 3D density distributions (but only in the case of line-of-sight velocities).

We assume the stars follow a tracer density $\rho(\boldsymbol{x},t)$ rotating at a constant pattern speed $\Omega_\mathrm{p}$. In terms of the Cartesian coordinates $(x,y,z)$ in the non-rotating disc frame, the continuity equation is given by
\begin{equation}
    \frac{\partial\rho}{\partial t} +\nabla\cdot(\rho\boldsymbol{v}) =\Omega_\mathrm{p}\Big[y\frac{\partial\rho}{\partial x}-x\frac{\partial\rho}{\partial y}\Big]+\nabla\cdot(\rho\boldsymbol{v})=0.
\label{eqn::continuity}
\end{equation}

Introducing the standard Galactic coordinates $(\ell,b)$ along with distance $s$ and corresponding transverse and line-of-sight velocities $v_\ell$, $v_b$ and $v_{||}$, the continuity equation becomes
\begin{equation}
\begin{split}
    &\Omega_\mathrm{p}\Big[-R_0\cos b\sin\ell\frac{\partial\rho}{\partial s}+\Big(1-\frac{R_0\cos\ell}{s\cos b}\Big)\frac{\partial\rho}{\partial\ell}
    +\frac{R_0}{s}\sin\ell\sin b\frac{\partial\rho}{\partial b}\Big]\\
    &+\frac{1}{s^2}\frac{\partial(s^2\rho v_{||})}{\partial s}+\frac{1}{s\cos b}\frac{\partial(\rho v_\ell)}{\partial \ell}+\frac{1}{s\cos b}\frac{\partial(\rho v_b \cos b)}{\partial b}=0.
\end{split}
\end{equation}
$R_0$ is the distance to the Galactic centre and the choice of coordinate systems is such that $\Omega_\mathrm{p}>0$ for the Galactic bar (i.e. a left-handed $(x,y,z)$ system). Here and in the subsequent expressions we have assumed the Sun is in the Galactic plane ($z=0$). In Appendix~\ref{Appendix:z0} we give the expressions incorporating the small offset ($\sim25\,\mathrm{pc}$) due to the Sun's height above the Galactic plane. Mirroring the method of \cite{TremaineWeinberg}, we multiply by $s^2\cos b$ and integrate with respect to $s$ from $0$ to $\infty$. We use the fact that $s^2\rho$ vanishes at $0$ and $\infty$ (provided $\rho$ falls off faster than $s^2$ at $\infty$ -- reasonable for a barred tracer) and integrate once by parts to write
\begin{equation}
\begin{split}
    &\Omega_\mathrm{p}\Big[2 R_0\cos^2 b\sin\ell\int_0^\infty\mathrm{d}s\,s\rho
    +\int_0^\infty\mathrm{d}s\,(s^2\cos b-
    R_0 s\cos\ell)\frac{\partial\rho}{\partial\ell}
    \\&
    +R_0\sin\ell\sin b\cos b\int_0^\infty\mathrm{d}s\,s\frac{\partial\rho}{\partial b}\Big]\\
    &+\int_0^\infty\mathrm{d}s\,s\frac{\partial(\rho v_\ell)}{\partial \ell}+\int_0^\infty\mathrm{d}s\,s\frac{\partial(\rho v_b \cos b)}{\partial b}=0.
\end{split}
\end{equation}
Noting $\partial/\partial b(\sin b\cos b)=2\cos^2b-1$, we can combine the $\ell$ and $b$-derivatives with the first integral such that
\begin{equation}
\begin{split}
    &\Omega_\mathrm{p}R_0\sin\ell\frac{\partial}{\partial b}\int_0^\infty\mathrm{d}s\,\rho s \sin b\cos b
     \\&+\Omega_\mathrm{p}\frac{\partial}{\partial\ell}\int_0^\infty\mathrm{d}s\,
     \rho s (s\cos b-
    R_0\cos\ell)
    \\
    &+\int_0^\infty\mathrm{d}s\,s\frac{\partial(\rho v_\ell)}{\partial \ell}+\int_0^\infty\mathrm{d}s\,s\frac{\partial(\rho v_b \cos b)}{\partial b}=0.
\end{split}
\label{eqn::lbderiv}
\end{equation}
We proceed by integrating in $b$ from $-\pi/2$ to $\pi/2$ imposing the condition that $\rho(b=\pm\pi/2)=0$ for all $s$ (this is valid provided the solar radius $R_0$ encompasses the entirety of the non-axisymmetry). This leaves only the second and third terms. A final integration in $\ell$ from $-\pi$ to $\ell$ (again requiring the non-axisymmetry to be completely encompassed such that $\rho(\ell=\pi)=0$ for all $s$) reduces the expression to
\begin{equation}
\begin{split}
    \Omega_\mathrm{p}(\ell) &= \frac{\int_{-\pi/2}^{\pi/2}\mathrm{d}b\int_0^\infty\mathrm{d}s\,s\rho v_\ell}{\int_{-\pi/2}^{\pi/2}\mathrm{d}b\int_0^\infty\mathrm{d}s\,
    s\rho(R_0 \cos\ell-s\cos b)}\\&\equiv\frac{\langle v_\ell \rangle_{b,s}}{\langle R_0\cos\ell-s\cos b\rangle_{b,s}},
    \end{split}
\end{equation}
where we have introduced the notation $\langle\rangle_{i,j}$ to denote multiplying by $s\rho$ and averaging over $i$ and $j$. Here we have written $\Omega_\mathrm{p}$ as a function of $\ell$ to make clear that the right-hand side is a function of $\ell$. However, provided the assumptions are satisfied, $\Omega_\mathrm{p}$ is a constant. Note we are free to multiply numerator and denominator by a general function $f(\ell)$ and integrate over $\ell$. This is the equivalent of \cite{TremaineWeinberg}'s $h(Y)$ which they recommend to be odd to minimise the effect of centering errors. For our application, it may be desirable to make $f(\ell)$ the inverse of the variance of the result for each $\ell$. We will see how we can combine the estimates probabilistically.

We observe that the derived expression is independent of the current bar angle $\alpha$. For a bar with no azimuthal extent oriented at $\alpha$, $\rho=|R|^{-1}\rho_R(|R|)\delta(\phi-\alpha)\rho_z(z)$, the Galactic longitude velocity $v_\ell(\ell)=R\Omega_\mathrm{p}\cos(\ell+\alpha)$ where $R$ is the cylindrical polar distance from the centre of the Galaxy to the intercept between the bar and the line-of-sight (taking negative sign for $\ell<0$). Therefore, $R\cos(\ell+\alpha)=R\cos\ell\cos\alpha-R\sin\ell\sin\alpha=R_0\cos\ell-s\cos b$ as $R_0=s\cos b\cos\ell+R\cos\alpha$ and $s\cos b\sin\ell=R\sin\alpha$. We recover the expected result.

We can return to equation~\eqref{eqn::lbderiv} and proceed in a similar fashion. We integrate instead in $\ell$ from $-\pi$ to $\pi$ and $b$ from $-\pi/2$ to $b$ to find
\begin{equation}
        \Omega_\mathrm{p}(b) = -\frac{\langle v_b\rangle_{\ell,s}}{
    R_0\sin b\langle \sin \ell\rangle_{\ell,s}}.
\end{equation}
The $\sin b$ in the denominator naturally makes this estimator poorly suited to measuring $\Omega_\mathrm{p}$ as any noise in $v_b$ is amplified for small $b$. The equivalent expression for the line-of-sight velocities is obtained by multiplying the continuity equation by $\cos b$, integrating over all $\ell$ and $b$ (integrating by parts once) and integrating $s$ from $\infty$ to $s$:
\begin{equation}
    \Omega_\mathrm{p}(s) = \frac{\langle v_{||}\cos b\rangle_{\ell,b}}{
    R_0\langle \sin\ell\cos^2 b\rangle_{\ell,b}}.
\end{equation}
This expression is given by \cite{Debattista2002} who proceed to multiply both numerator and denominator by a general function $f(s)$ and integrate over all distance, $s$. Note that our three estimators have all reduced the 3D continuity equation to considering flows in a single dimension by integrating over the other two coordinates. Each expression in essence measures the rotational velocity divided by the distance to the Galactic centre. However, they are not exactly equivalent. This is because a bar is preferentially supported by prograde orbits. As \cite{Qin2015} showed, simply using the rotational velocity at a few spatial locations to estimate the pattern speed leads to biases. The geometric factors and averaging in the estimators are a necessary component.

These estimators are valid provided (i) the pattern is steady, (ii) the integration encompasses the entirety of the non-axisymmetry and (iii) there is a single pattern speed. It is likely that the pattern speed of the bar is evolving slowly -- the impact of this can be tested using our reference simulation. The validity of the second and third assumptions is less clear, particularly in the presence of other non-axisymmetries such as spiral arms. Our modelling extracts non-axisymmetries only in the red giant populations which do not trace recent star formation and so are likely free from small-scale non-axisymmetries. It is likely that the entire bar rotates at a single pattern speed as differentially rotating triaxial structures rapidly exchange angular momentum.

\subsection{Using heliocentric velocities}

Our expressions were derived using velocities measured in the Galactic rest frame. For heliocentric velocities (denoted by primes), we must first correct for the solar reflex motion as
\begin{equation}
    \begin{split}
        v_{||}&=v'_{||}+u_\odot\cos\ell\cos b+v_\odot\sin\ell\cos b+w_\odot\sin b,\\
        v_{\ell}&=v'_{\ell}-u_\odot\sin\ell+v_\odot\cos\ell,\\
        v_{b}&=v'_{b}-u_\odot\cos\ell\sin b-v_\odot\sin\ell\sin b+w_\odot\cos b,
    \end{split}
\end{equation}
where $(u_\odot,v_\odot,w_\odot)$ is the solar velocity in the Galactic rest frame (positive $u$ towards the centre of the Galaxy and positive $v$ in the direction of Galactic rotation). Our pattern speed expressions in terms of heliocentric coordinates are then given by
\begin{equation}
    \Omega_\mathrm{p}(s)=\frac{v_\odot}{R_0}+ \frac{ \langle(v'_{||}+u_\odot\cos\ell\cos b+w_\odot\sin b)\cos b\rangle_{\ell,b}}{\langle
    R_0 \sin\ell\cos^2 b\rangle_{\ell,b}}.
\end{equation}
\begin{equation}
    \Omega_\mathrm{p}(b)=\frac{v_\odot}{R_0}- \frac{\langle v'_{b}-u_\odot\cos\ell\sin b+w_\odot\cos b\rangle_{\ell,s}}{\langle
    R_0\sin\ell\sin b\rangle_{\ell,s}}.
\end{equation}
\begin{equation}
    \Omega_\mathrm{p}(\ell)=\frac{v_\odot-u_\odot\tan\ell+\mathcal{F}}{R_0-\mathcal{K}}
    ,\>
    \mathcal{F}=\frac{\langle v'_\ell\rangle_{b,s}}{\langle \cos\ell\rangle_{b,s}},\>\mathcal{K}=\frac{\langle s\cos b\rangle_{b,s}}{\langle\cos\ell\rangle_{b,s}}
    \label{eqn::fk_equation}
\end{equation}
Note that for $\Omega_\mathrm{p}(s)$ and $\Omega_\mathrm{p}(b)$, the pattern speed is completely degenerate with $v_\odot/R_0$ for all $\ell,b,s$, whilst for $\Omega_\mathrm{p}(\ell)$ the degeneracy is more complex and depends on the geometric factor $\mathcal{K}$. In the thin bar limit, $\mathcal{K}=R_0$ when the viewing angle is orthogonal to the bar as in this case $v_\ell$ contains no contribution from the azimuthal rotation.

The impact of the solar velocity depends on the estimator employed. $u_\odot$ and $w_\odot$ are well measured as $u_\odot=(11.1\pm0.7\pm1\,\mathrm{sys}.)\,\mathrm{km\,s^{-1}}$ and $w_\odot=(7.25\pm0.36\pm0.50\,\mathrm{sys}.)\,\mathrm{km\,s^{-1}}$ \citep[][although the local standard of rest may have a non-zero $u$ velocity]{Schoenrich2010}. Uncertainties in $u_\odot$ primarily affect $\Omega_\mathrm{p}(s)$ as $\Omega_\mathrm{p}(\ell)$ contains $u_\odot\tan\ell$ (and $|\ell|<10\,\mathrm{deg}$). Assuming Sgr A* is a rest with respect to the bulge, $v_\odot/R_0$ is well constrained from the proper motion of Sgr A* as $\mu_{\ell,A*}=-v_\odot/(4.74 R_0)=(-6.379\pm0.026)\,\mathrm{mas\,yr}^{-1}$ \citep{Reid2004}. Therefore, we conclude that the uncertainty in the pattern speed estimators, $\Omega_\mathrm{p}(s)$ and $\Omega_\mathrm{p}(b)$, comes primarily from the unknown Galactic centre distance $R_0$. Fixing $\mu_{\ell,A*}$ and ignoring the impact of $u_\odot$ and $w_\odot$, we have that both $\Omega_\mathrm{p}(s)$ and $\Omega_\mathrm{p}(b)$ fall with $R_0$ like $\Omega_\mathrm{p}=k_1+k_2/R_0$ whilst for $\Omega_\mathrm{p}(\ell)$ we find
\begin{equation}
    \Omega_\mathrm{p}(\ell)=\frac{-4.74\mu_{\ell,A*}R_0+\mathcal{F}}{R_0-\mathcal{K}}.
\end{equation}
This more complex behaviour allows the possibility of measuring $R_0$ and $\Omega_\mathrm{p}$ from the data. However, with $\mu_{\ell,A*}$ fixed, inference of $R_0$ is degenerate with a fractional distance systematic (i.e. a constant absolute magnitude offset). The combination of both $\Omega_\mathrm{p}(s)$ and $\Omega_\mathrm{p}(\ell)$ puts stronger restrictions on $R_0$, $v_\odot$, $u_\odot$ and $\Omega_\mathrm{p}$, potentially testing the assumption of steady state. However, such an analysis would require a spectroscopic dataset distributed over all $b$ and $\ell$.

\subsection{Estimators from Jeans' equations}

We briefly discuss the possibility of using higher order equations derived from the collisionless Boltzmann equation (CBE) to measure the pattern speed. These could provide tighter constraints on the pattern speed. After the continuity equation, the next two equations (obtained by multiplying the CBE by $v_j$ and $v_jv_k$ respectively and integrating over all velocities) are
\begin{equation}
\begin{split}
\frac{\partial(\rho\overline{v}_j)}{\partial t}+\frac{\partial(\rho\overline{v_iv_j})}{\partial x_i}+\rho\frac{\partial\Phi}{\partial x_j}&=0,\\
\frac{\partial(\rho\overline{v_jv_k})}{\partial t}+\frac{\partial(\rho\overline{v_iv_jv_k})}{\partial x_i}+\overline{v}_j\rho\frac{\partial\Phi}{\partial x_k}+\overline{v}_k\rho\frac{\partial\Phi}{\partial x_j}&=0,
\end{split}
\label{eqn::higherorder}
\end{equation}
where we have introduced the potential $\Phi$. Using the first of these equations combined with the continuity equation and integrating over all space \citep[c.f.][]{BinneyTremaine2008}, we find
the tensor-virial theorem in the form \citep{ChandrasekharEFE}
\begin{equation}
2K_{ij}+W_{ij}+\Omega_\mathrm{p}^2(J_{ij}-\delta_{3i}J_{3j})+2\levicivita{i}{k}{3}\Omega_\mathrm{p}\int_V\mathrm{d}^3\boldsymbol{x}\,\rho w_k x_j=0,
\end{equation}
where the velocities are in the rotating frame $w_i=v_i-\levicivita{i}{3}{k}\Omega_\mathrm{p}x_k$ and the tensors are
\begin{equation}
    K_{ij}=\int_V\mathrm{d}^3\boldsymbol{x}\rho w_i w_j,\>\>
    W_{ij}=\int_V\mathrm{d}^3\boldsymbol{x}\rho x_i a_j,\>\>
    J_{ij}=\int_V\mathrm{d}^3\boldsymbol{x}\rho x_i x_j.
\end{equation}
Imposing triaxial symmetry removes the final Coriolis-like term leaving a balance between random kinetic energy, potential energy and the rotational energy of the figure. Also, all off-diagonal relations are zero. We are then left with three relations between the pattern speed and the three unknown potential energy tensor diagonal components. We therefore cannot use these relations to infer the pattern speed without  more knowledge of the potential (e.g., that the density is ellipsoidally stratified).

Another approach is to combine the two expressions in equation~\eqref{eqn::higherorder} to eliminate the potential derivatives \citep{KuijkenTremaine1991}
\begin{equation}
\frac{\partial\sigma_{jk}^2}{\partial t}+\sigma^2_{ik}\frac{\partial\overline{v}_j}{\partial x_i}+\sigma^2_{ij}\frac{\partial\overline{v}_k}{\partial x_i}+\overline{v}_i\frac{\partial\sigma^2_{jk}}{\partial x_i}
+\frac{1}{\rho}\frac{\partial(\rho\sigma^3_{ijk})}{\partial x_i}=0,
\end{equation}
where $\sigma^2_{ij}\equiv\overline{(v_i-\overline{v_i})(v_j-\overline{v_j})}$ and $\sigma^3_{ijk}\equiv\overline{(v_i-\overline{v_i})(v_j-\overline{v_j})(v_k-\overline{v_k})}$. If we neglect the third order moments $\sigma_{ijk}^3$ then this equation gives a simple conservation law for the kinetic pressure. Assuming a constant pattern speed, we can perform the same replacement giving
\begin{equation}
\Omega_\mathrm{p}\Big[y\frac{\partial\sigma^2_{jk}}{\partial x}-x\frac{\partial\sigma^2_{jk}}{\partial y}\Big]+\sigma^2_{ik}\frac{\partial\overline{v}_j}{\partial x_i}+\sigma^2_{ij}\frac{\partial\overline{v}_k}{\partial x_i}+\overline{v}_i\frac{\partial\sigma^2_{jk}}{\partial x_i}=0.
\end{equation}
However, here we observe that the right-hand side involves all three velocity components and all three components of $\sigma^2_{ij}$ (for fixed $j$). This is not a limitation in the Tremaine-Weinberg method where terms involving unknown velocities are integrated out. However, we cannot write the terms on the right-hand side of this equation purely as sums of derivatives making replication of the Tremaine-Weinberg method hard. With a full knowledge of the velocity field, this equation (with or without the inclusion of the $\sigma_{ijk}^3$ term) offers an attractive way to measure the pattern speed from the dispersion (and mean velocity) field without knowledge of the potential.

\subsection{Probablistic approach}

We develop a framework for the application of the estimator $\Omega_\mathrm{p}(\ell)$ provided in the previous section that accounts for uncertainty in the observables as well as the Galactic parameters. Such an approach is necessary for a robust estimate of $\Omega_\mathrm{p}$ with associated uncertainties. The problem is essentially equivalent to a linear regression with correlated uncertainties in both `$x$' and `$y$' \citep[e.g.][]{Me95} and a prior on the intercept.

We consider as our measurements at each $\ell$ the `vector' $\boldsymbol{\mathcal{X}}_i=(\mathcal{F},\mathcal{K})_i$ with corresponding covariance $\boldsymbol{\Sigma}_{\mathcal{X}i}$. Both the numerator and denominator of $\Omega_\mathrm{p}(\ell)$ tend to zero for $\ell\rightarrow0$ i.e. the centre of the bar/bulge is not translating with respect to the Galaxy and the centre is located at $R_0$. Therefore, both numerator and denominator can be modelled by a power series with no constant coefficient:
\begin{equation}
f(\ell)=v_\odot-u_\odot\tan\ell+\mathcal{F}=\Omega_\mathrm{p}(R_0-\mathcal{K})=\Omega_\mathrm{p}R_0\sum^{n=N_\mathrm{max}}_{n=1}c_n(\tan\ell)^n.
\end{equation}
We expand in terms of $\tan\ell$ as for a needle-thin bar at angle $\alpha$ $R_0-\mathcal{K}=R_0\tan\ell/\tan(\ell+\alpha)$. Inspecting our reference simulation (see Section~\ref{Section::Recovery}), we find that $1\lesssim N_\mathrm{max}\lesssim3$ is appropriate, so we set $N_\mathrm{max}=3$ for application to the data. This approach is chosen rather than modelling $\Omega_\mathrm{p}(\ell)$ directly as the uncertainty in $\Omega_\mathrm{p}(\ell)$ becomes larger as $\ell\rightarrow0$ whilst $\mathcal{F}$ and $\mathcal{K}$ are well behaved.

We construct the likelihood
\begin{equation}
\begin{split}
    \boldsymbol{\mathcal{X}}_i&\sim\mathcal{N}(\boldsymbol{f}(\ell_i),\boldsymbol{\Sigma}_{\mathcal{X}i}+\boldsymbol{\Sigma}_f),\\
    \boldsymbol{f}(\ell_i)&=\lambda\Big( f(\ell_i)-v_\odot+u_\odot\tan\ell_i,R_0-\frac{ f(\ell_i)}{\Omega_\mathrm{p}}\Big),\\ \boldsymbol{\Sigma}_f&=\mathrm{diag}(\sigma^2_\mathcal{F},\sigma^2_\mathcal{K}),
\end{split}
\label{Eqn::OmegaModel}
\end{equation}
where $\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ is a multidimensional normal distribution with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. The diagonal covariance matrix $\boldsymbol{\Sigma}_f$ represents an intrinsic scatter in $\mathcal{F}$ and $\mathcal{K}$ which could arise from the neglected boundary terms. $\lambda$ is a fractional distance systematic, so all observed distances are a factor $\lambda$ too large. We adopt the following priors:
\begin{equation}
    \begin{split}
        R_0/\,\mathrm{kpc}&\sim\mathcal{N}(\mu_{R0},\sigma_{R0}),\\
        \mu_{\ell,A*}/\,\mathrm{mas\,yr}^{-1}&\sim\mathcal{N}(-6.379,0.026),\\
        \lambda&\sim\mathcal{N}(1,0.05),\\
        u/\,\mathrm{km\,s}^{-1}&\sim\mathcal{N}(11.1,1.2),\\
        \ln\sigma_\mathcal{F}/\mathrm{km\,s}^{-1}&\sim\mathcal{U}(-10,10),\\
        \ln\sigma_\mathcal{K}/\mathrm{kpc}&\sim\mathcal{U}(-10,10),
    \end{split}
\end{equation}
where $\mathcal{N}(\mu,\sigma)$ is a normal with mean $\mu$ and standard deviation $\sigma$ and $\mathcal{U}(a,b)$ is uniform between $a$ and $b$.
For $R_0$ we consider three priors: $\mathcal{N}(8.12\,\mathrm{kpc},0.03\,\mathrm{kpc})$ from \cite{GravityCollab2018}, $\mathcal{N}(8.2\,\mathrm{kpc},0.09\,\mathrm{kpc})$ from \cite{McMillan2017} (where it should be noted that \cite{McMillan2017} uses the proper motion of Sgr A* in the estimate of $R_0$ leading to a covariance between $v_\odot$ and $R_0$) and $\mathcal{N}(8.12\,\mathrm{kpc},2\,\mathrm{kpc})$. The second prior is approximately that suggested by \cite{BlandHawthornGerhard2016} from combining estimates from multiple studies. The third prior is an uninformative prior. We also consider an uninformative prior on $\mu_{\ell,A*}$ of $\mathcal{N}(-6.379\,\mathrm{mas\,yr}^{-1},1\,\mathrm{mas\,yr}^{-1})$. Our prior on $\lambda$ corresponds approximately to considering magnitude systematics of $\sim0.1\,\mathrm{mag}$. We write the model in Stan and sample using the NUTS sampler \citep{Hoffman2011}.

\section{Application to data}\label{Section::Application}

We apply the derived framework to the data produced in Paper I. In that work, we derived the mean transverse velocities $v_\ell$ and $v_b$ and corresponding uncertainties as a function of distance across the bar/bulge region from $\sim45$ million giant stars in VVV. We used a combination of proper motions from Gaia DR2 \citep{Gaia2018} and VIRAC v1.1 \citep{Sm18}, an astrometric catalogue derived from the VVV observations calibrated absolutely using Gaia DR2. In small fields of $0.2\,\mathrm{deg}\times0.2\,\mathrm{deg}$ across the VVV bulge footprint, the bulge giants (both red giant branch and red clump) were selected in a unextincted colour-magnitude box of $0.4<(J-K_s)_0<1$ and $11.5<K_{s0}<14.5$ using a two-dimensional extinction map from the method of \cite{Gonzalez2011}. First, the density in each field was measured assuming a luminosity function \citep[taken from][]{Simion2017} and accounting for incompleteness, and then the density distribution was used to extract the first two moments of the transverse velocity distributions as a function of distance by probabilistically considering the possible velocities for each star given its $K_{s0}$ magnitude. Here we utilise the mean longitudinal transverse velocities with location $v_\ell(\ell,b,s)$ and the density field $\rho(\ell,b,s)$ computed on a grid in log(distance), $\ell$ and $b$. The integrals are computed using the trapezoidal rule but using Simpson's rule instead gives very similar results. We fill in the region $5<b/\,\mathrm{deg}<10$ using $-10<b/\,\mathrm{deg}<-5$ assuming reflection symmetry in $b=0$. This increases the estimates of the pattern speed by $\sim2\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$. The uncertainties $\boldsymbol{\Sigma}_{\boldsymbol{\mathcal{X}}_i}$ are computed through propagation of uncertainties in density and velocity. We recall that due to the procedure in Paper I the uncertainties in the velocities are probably underestimated as they only account for proper motion uncertainties not uncertainties in the density (or luminosity function) which propagate to increased uncertainties in the physical velocities.

In Table~\ref{Table::results}, we show the results of applying the method to different subsets of the data and the different priors. We report the estimates of $R_0$, $\mu_{\ell,A*}$ and $\Omega_\mathrm{p}$ as well as the chi-squared per datapoint which is given by
\begin{equation}
    \chi^2/N \equiv \frac{1}{N}\sum^N_i\frac{(\Omega_\mathrm{p}-\Omega_\mathrm{p}(\ell_i))^2}{\sigma_\Omega^2(\ell_i)},
\end{equation}
where $\Omega_\mathrm{p}$ is the median from the MCMC chain, $\Omega_\mathrm{p}(\ell_i)$ the estimate for bin $i$ and $\sigma_\Omega(\ell_i)$ the associated uncertainty (from propagating the uncertainties in $\mathcal{F}$ and $\mathcal{K}$).

We see that when using all of the data with the `tight' prior we obtain $\Omega_\mathrm{p}=(37.2\pm3.3)\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$, but the $\chi^2/N=19.7$ is poor and $\mu_{\ell,A*}$ is $2\sigma$ away from the prior. Using the 'loose' prior, we find the pattern speed remains similar, the $\chi^2/N=12.3$ reduces but still $\mu_{\ell,A*}$ is in tension. Relaxing the $R_0$ prior further only weakly improves $\chi^2/N$ but $R_0$ does not stray significantly. Using a weak prior on $\mu_{\ell,A*}$, we see the data `wants' to reduce $|\mu_{\ell,A*}|$ producing a low pattern speed of $\Omega_\mathrm{p}=(23.7\pm2.7)\,\mathrm{km\,s}^{-1}$ but not with a significantly improved $\chi^2/N$.

We next have tried separating the data into $\ell>0$ and $\ell<0$. The two models with lowest $\chi^2/N$ are $\ell<0$ `No $R_0$' with  $\Omega_\mathrm{p}=(34.7\pm5)\,\mathrm{km\,s}^{-1}$ but producing an inconsistent $R_0$ of $(9.42\pm0.45)\,\mathrm{kpc}$ and $\ell>0$ `No $R_0$' with $\Omega_\mathrm{p}=(30.9\pm1.2)\,\mathrm{km\,s}^{-1}$ but producing an inconsistent $\mu_{\ell,A*}$ of $(-6.24\pm0.01)\,\mathrm{mas\,yr}^{-1}$. We next tried removing the central regions $|\ell|>2.5\,\mathrm{deg}$ as these produce the noisiest estimates of $\Omega_\mathrm{p}$ as $v_\ell$ is small. The set of models for $\ell>2.5\,\mathrm{deg}$ produce very satisfactory results with low $\chi^2/N\approx1.3-1.5$ and both $R_0$ and $\mu_{\ell,A*}$ consistent with expectations both with and without the prior. All models give $\Omega_\mathrm{p}\approx(41\pm3)\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$. For $\ell<-2.5\,\mathrm{deg}$ the situation is less satisfactory with higher $\chi^2/N$. The tension is illustrated by the `No $R_0$' model which produces a much too high estimate of $R_0$ but $\Omega_\mathrm{p}$ consistent with $\ell>2.5\,\mathrm{deg}$. When combining both positive and negative $\ell$, we obtain $\Omega_\mathrm{p}\approx(30\pm1)\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ but the $\chi^2/N$ are higher and we always recover $\mu_{\ell,A*}$ many $\sigma$ away from the measured value.

We can understand these results by inspecting Fig.~\ref{fig:fk_balance} which shows the two terms in equation~(\ref{eqn::fk_equation}) that must be equal for the pattern speed to be constant across the bar. We see a pattern speed of $\Omega_\mathrm{p}=40\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ produces a consistent result for $\ell>0$ but the magnitude of the velocities is too small (or the distances too small) for $\ell<0$. This is fixed by adopting $\Omega_\mathrm{p}=25\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ or $\mu_{\ell,A*}=-6.15\,\mathrm{mas\,yr}^{-1}$ (many $\sigma$ from the measured value) but produces a poorer fit for $\ell>0$. Note the grey region $|\ell|<2.5\,\mathrm{deg}$ where the estimator is noisy.

An alternative way of viewing the data is plotting $\mathcal{F}-u_\odot\tan\ell$ against $\mathcal{K}$ as shown in Fig.~\ref{fig:fk_gradient}. The gradient in this plane gives the pattern speed. We observe how the points with $|\ell|<2.5\,\mathrm{deg}$ are significantly deviant any straight line fit validating their removal. We see that the near-side of the bar produces a linear fit with gradient $\Omega_\mathrm{p}=41\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ and intercept consistent with the Galactic centre position and velocity. However, inclusion of the far-side reduces the gradient to $\Omega_\mathrm{p}=31\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ but produces an inconsistent intercept.
\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/F_K_balance.pdf}
    \caption{Terms in our pattern speed estimator $(R_0-\mathcal{K})$ (black using $R_0=8.12\,\mathrm{kpc}$) and $(-4.74\mu_{\ell,A*}R_0-u_\odot\tan\ell+\mathcal{F})/\Omega_\mathrm{p}$ (coloured) in units of $\mathrm{kpc}$ and binned in $\ell$. When the two terms are identical, the pattern speed is constant and correct across the bar. The blue dashed line uses $\mu_{\ell,A*}=-6.379\,\mathrm{mas\,yr}^{-1}$ and $\Omega_\mathrm{p}=40\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$, green dotted reduces $\Omega_\mathrm{p}=25\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ and red dash-dot increases $\mu_{\ell,A*}=-6.15\,\mathrm{mas\,yr}^{-1}$. We see the far-side of the bar requires a lower pattern speed or higher $\mu_{\ell,A*}$. Within the shaded region $|\ell|<2.5\,\mathrm{deg}$ the estimator is noisy.}
    \label{fig:fk_balance}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/fk_colorl.pdf}
    \caption{Terms in our pattern speed estimator $\mathcal{K}$ vs. $-u_\odot\tan\ell+\mathcal{F}$ for each $\ell$ bin (given by the colour). The gradient of this plot corresponds to the pattern speed. The points with black edges have $|\ell|<2.5\,\mathrm{deg}$ and clearly lie away from the trend. The red cross shows the Galactic centre (at $R_0=8.12\,\mathrm{kpc}$ and $-4.74\mu_{\ell,A*}R_0=245.5\,\mathrm{km\,s}^{-1}$). The black solid line shows approximately our best fit to $\ell>2.5\,\mathrm{deg}$ with $\Omega_\mathrm{p}=41\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ whilst the dashed line shows the best fit to the entire data with $\Omega_\mathrm{p}=31\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ which doesn't pass through the Galactic centre.}
    \label{fig:fk_gradient}
\end{figure}

In Fig.~\ref{fig:data_application}, we show the results from our most successful model $\ell>2.5\,\mathrm{deg}$ with the `tight' prior. We see the posteriors approximately follow the priors for $R_0$, $\mu_{\ell,A*}$ and $\lambda$. The residuals in $\Omega_\mathrm{p}$ with respect to the model clearly have small-scale systematic variations and we see for $\ell<3.5\,\mathrm{deg}$ the pattern speed is biased high.

It is puzzling why $\ell<0$ produces poor results. One reason could be that we do not have sufficient coverage in distance. It appears the pattern speed is increased if the nearby data (distance less than $6\,\mathrm{kpc}$) is cut out. This hints that there is insufficient background disc to counteract the foreground disc which is biasing the signal. We have experimented with using a fixed distance range of data relative to the bar major axis, but no one distance cut can reliably be chosen over any other. A further concern is unreliable extinction estimates which bias the velocities via incorrect distance estimates. We are using a 2d extinction map which is poor near the plane. For $\ell\ll0$ the bar gets closer to the plane due to geometric effects making this a bigger problem than for $\ell>0$. In the bottom section of Table~\ref{Table::results} we have removed $|b|<1\,\mathrm{deg}$ which produces very similar results -- a pattern speed of $\sim31\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ and a low $-\mu_{\ell,A*}$ -- although the pattern speed is slightly higher (see later simulation tests).
Furthermore, in the analysis a proper motion systematic is degenerate with the proper motion of Sgr A*. In Fig.~\ref{fig:meanl} we show the mean proper motion averaged along the line-of-sight (accounting for the incompleteness effects). We see the pattern of the Gaia scanning law and associated systematic variations of the mean proper motion of up to $0.5-1\,\mathrm{mas\,yr}^{-1}$. These systematics are particularly bad for $\ell<-2.5\,\mathrm{deg}$ but appear less severe in the region $\ell>2.5\,\mathrm{deg}$ where our estimator is performing better. We have tested that this systematic pattern is present irrespective of using Gaia data in addition to VIRAC and irrespective of different cuts on proper motion quality. It is also independent of magnitude suggesting it arises from the relative-to-absolute correction for the VIRAC v1.1 (see Paper I).

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/data_corner_plot.pdf}
    \caption{Posterior distributions for the model parameters when applied to data with $\ell>2.5\,\mathrm{deg}$. The grey dotted lines show the assumed prior. The top right inset shows the pattern speed estimate at each $\ell$ along with the median and $1\sigma$ error band for the pattern speed. $\lambda$ is a fractional distance systematic.}
    \label{fig:data_application}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/mean_l_systematics.pdf}
    \includegraphics[width=\columnwidth]{figures/mean_vl_field.pdf}
    \caption{On-sky line-of-sight averaged mean $\mu_\ell$ (weighted by distance pdf) and mean $\ell$ velocity relative to Sgr A* (weighted by $s\rho$ as in the pattern speed estimator). Near the plane for $\ell<-2.5\,\mathrm{deg}$, the Gaia scanning law is clearly visible giving rise to proper motion systematics of $\sim0.5-1\,\mathrm{mas\,yr}^{-1}$. The grey shaded region shows $|\ell|<2.5\,\mathrm{deg}$ which we ignore in some of our fits.}
    \label{fig:meanl}
\end{figure}

In conclusion, our `best' model as measured by $\chi^2/N$ is for $\ell>2.5\,\mathrm{deg}$ with any choice of prior (all produce similar results). This yields a pattern speed of $\Omega_\mathrm{p}=(41\pm3)\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$. Assuming a $v$ peculiar motion of the Sun of $12.24\,\mathrm{km\,s}^{-1}$ \citep{Schoenrich2010} and no non-axisymmetric streaming in the solar neighbourhood, this yields a corotation radius of
$(5.7\pm0.4)\,\mathrm{kpc}$.
We find an identical result if we instead use the rotation curve from \cite{Eilers2019}.

Our errorbars here are formal and likely underestimate the true uncertainty, particularly as systematic issues haven't enabled a consensus on the pattern speed to be formed across the observed volume and we know the uncertainties in the mean velocities are underestimated as we have not fully propagated the uncertainty in the density field or in the luminosity function. In Appendix~\ref{appendix:lf} we discuss our choice of luminosity function and demonstrate a broader red clump magnitude distribution ($0.12\,\mathrm{mag}$ instead of $0.067\,\mathrm{mag}$) gives near identical results.

\begin{table*}
    \caption{Results of application of our method to the data. The first column describes the subset of data used. Second column the priors employed (`Tight' is $R_0=(8.12\pm0.03)\,\mathrm{kpc}$ and $\mu_{\ell,A*}=(-6.379\pm0.026)\,\mathrm{mas\,yr}^{-1}$, `Loose' is $R_0=(8.2\pm0.09)\,\mathrm{kpc}$ and $\mu_{\ell,A*}=(-6.379\pm0.026)\,\mathrm{mas\,yr}^{-1}$, `No $R_0$' is $R_0=(8.12\pm2)\,\mathrm{kpc}$ and $\mu_{\ell,A*}=(-6.379\pm0.026)\,\mathrm{mas\,yr}^{-1}$ and `No $\mu_{\ell,A*}$' is $R_0=(8.12\pm0.03)\,\mathrm{kpc}$ and $\mu_{\ell,A*}=(-6.379\pm1)\,\mathrm{mas\,yr}^{-1}$. Subsequent columns give the median and standard deviation of the parameters. The final column gives the chi-squared per $\ell$ bin which is an approximate reflection of the goodness of fit.}
    \centering
    \begin{tabular}{c|c|c|c|c|c}
         Subset & Prior & $R_0/,\mathrm{kpc}$ & $-
         \mu_{\ell,A*}/\,\mathrm{mas\,yr}^{-1},$ & $\Omega_\mathrm{p}/\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$ & $\chi^2/N$ \\
         \hline
All&Tight&$8.12\pm0.03$&$6.30\pm0.03$&$37.20\pm3.33$&$19.70$\\
&Loose&$8.21\pm0.08$&$6.31\pm0.03$&$36.85\pm3.09$&$12.32$\\
&No $R_0$&$8.24\pm0.14$&$6.30\pm0.03$&$37.07\pm3.21$&$11.36$\\
&No $\mu_{\ell,A*}$&$8.14\pm0.03$&$6.13\pm0.04$&$23.75\pm2.71$&$13.64$\\
\\
$\ell<0$&Tight&$8.12\pm0.03$&$6.29\pm0.03$&$27.08\pm1.96$&$19.58$\\
&Loose&$8.25\pm0.08$&$6.36\pm0.02$&$22.87\pm1.92$&$19.11$\\
&No $R_0$&$9.42\pm0.45$&$6.35\pm0.03$&$34.65\pm5.16$&$6.45$\\
&No $\mu_{\ell,A*}$&$8.13\pm0.03$&$6.31\pm0.05$&$26.47\pm3.03$&$20.83$\\
\\
$\ell>0$&Tight&$8.13\pm0.03$&$6.24\pm0.01$&$30.77\pm1.16$&$13.52$\\
&Loose&$8.22\pm0.06$&$6.24\pm0.01$&$30.81\pm1.20$&$8.93$\\
&No $R_0$&$8.24\pm0.08$&$6.24\pm0.01$&$30.88\pm1.18$&$7.31$\\
&No $\mu_{\ell,A*}$&$8.13\pm0.03$&$6.20\pm0.01$&$30.94\pm1.11$&$11.41$\\
\\
$\ell>2.5$&Tight&$8.12\pm0.03$&$6.36\pm0.02$&$42.09\pm2.50$&$1.42$\\
&Loose&$8.19\pm0.08$&$6.37\pm0.02$&$41.17\pm2.60$&$1.37$\\
&No $R_0$&$8.18\pm0.15$&$6.36\pm0.02$&$41.25\pm2.93$&$1.34$\\
&No $\mu_{\ell,A*}$&$8.12\pm0.03$&$6.32\pm0.04$&$39.41\pm3.63$&$1.51$\\
\\
$\ell<-2.5$&Tight&$8.13\pm0.03$&$6.37\pm0.02$&$23.26\pm1.75$&$3.44$\\
&Loose&$8.25\pm0.08$&$6.36\pm0.02$&$22.87\pm1.92$&$3.74$\\
&No $R_0$&$9.42\pm0.45$&$6.35\pm0.03$&$34.65\pm5.16$&$2.63$\\
&No $\mu_{\ell,A*}$&$8.13\pm0.03$&$6.31\pm0.05$&$26.47\pm3.03$&$2.60$\\
\\
$|\ell|>2.5$&Tight&$8.14\pm0.03$&$6.26\pm0.01$&$30.92\pm0.82$&$2.98$\\
&Loose&$8.24\pm0.06$&$6.26\pm0.01$&$30.96\pm0.78$&$2.70$\\
&No $R_0$&$8.28\pm0.08$&$6.26\pm0.01$&$31.00\pm0.77$&$2.58$\\
&No $\mu_{\ell,A*}$&$8.14\pm0.03$&$6.24\pm0.01$&$30.86\pm0.72$&$2.24$\\
\\
$|\ell|>2.5,|b|>1$&Tight&$8.13\pm0.03$&$6.21\pm0.01$&$31.70\pm0.69$&$4.13$\\
&Loose&$8.22\pm0.06$&$6.21\pm0.01$&$31.70\pm0.72$&$4.03$\\
&No $R_0$&$8.24\pm0.08$&$6.21\pm0.01$&$31.76\pm0.68$&$4.11$\\
&No $\mu_{\ell,A*}$&$8.13\pm0.03$&$6.19\pm0.01$&$31.29\pm0.64$&$3.81$\\

    \end{tabular}
    \label{Table::results}
\end{table*}

\section{Recovery of the pattern speed from a simulation}\label{Section::Recovery}
We provide a series of tests of our method for measuring the pattern speed from proper motion data using the continuity equation by application to a simulation. We first describe the simulation considered.

\subsection{Reference simulation}\label{Section::Simulation}

For the interpretation and testing of our results, we construct a simple reference simulation of a barred galaxy. This simulation is designed to approximately match the properties of the Milky Way although not to the level of detail of a full Made-to-Measure model \citep{Portail2017}. We use the initial condition generation \texttt{mkgalaxy} from \cite{McMillanDehnen2007}. The galaxy has three components: a disc, a bulge and a dark halo. We use the standard parameters: a \cite{Dehnen1999} disc with scale-length $R_d=1$, scale-height $z_d=0.1$ and mass $M_d=1$, a spherical \cite{Hernquist1990} bulge with scale-length $R_b=0.2$ and mass $M_b=0.2$, and a spherical \cite{NFW} halo with scale-length $R_h=6$ and mass $M_h=24$. The disc contains $200,000$ particles, the bulge $40,000$ and the halo $1,200,000$. The disc has a Toomre $Q$ parameter of $1.2$ making it radially-unstable to bar formation. Upon evolution for $200$ time units with \texttt{gyrfalcON} \citep{Dehnen2000}, the disc rapidly forms a bar that slows to $\Omega_\mathrm{p}\approx0.4$ by the end of the simulation. We measure the pattern speed from consecutive snapshots using the angular velocity of the second axis of the moment of inertia tensor for particles within $1.5$ simulation units of the centre. Using all particles and only those between $0.2$ and $0.5$ simulation units away from the midplane gives very similar results after $70$ simulation time units. We scale the final snapshot such that the scale-length of the disc is $2.5\,\mathrm{kpc}$ and the circular velocity of the disc $230\,\mathrm{km\,s}^{-1}$. This produces a bar rotating with pattern speed $\Omega_\mathrm{p}=45\,\mathrm{km\,s^{-1}\,kpc^{-1}}$ which we view at an angle of $\sim33\,\mathrm{deg}$ relative to the major axis. In Fig.~\ref{fig:simulation_diagram} we show two consecutive snapshots from the simulation scaled to the Milky Way.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/sim_diagram.pdf}
    \caption{Top-down view of the reference simulation at two consecutive snapshots. The displayed particles have $|z|<2\,\mathrm{kpc}$. Black lines delineate the VVV footprint and the blue line shows the major axis of the bar. The two snapshots are $\sim8\,\mathrm{Myr}$ apart.}
    \label{fig:simulation_diagram}
\end{figure}

\subsection{Application}\label{Section::Simulation_Application}

We consider particles $-10<\ell/\,\mathrm{deg}<10$, $-10<b/\,\mathrm{deg}<5$, $-6<(s-8.12)/\,\mathrm{kpc}<6$ for a solar location of $R_0=8.12\,\mathrm{kpc}$ viewing the bar at $\sim33\,\mathrm{deg}$. Proper motions are computed using the total solar velocity of $(u,v,w)_\odot=(11.1,-4.74\mu_{\ell,A*}R_0,7.25)\,\mathrm{km\,s}^{-1}$ and $\mu_{\ell,A*}=-6.379\,\mathrm{mas\,yr}^{-1}$.

When testing with simulations, we require expressions equivalent to those in Section~\ref{Section::PatternSpeed} but appropriate for a finite sampling of the underlying smooth functions. For a set of tracer particles, $\rho=\sum_im_i\delta^{(3)}(\boldsymbol{x}-\boldsymbol{x}_i)$ we construct bins (indexed by $n$) in $\ell$, $b$ or $s$ depending on the free variable in the pattern speed expression. The bins are centred on e.g. $\ell_n$ with width $\Delta\ell$. We evaluate
\begin{equation}
    \Omega_\mathrm{p}(\ell_n) = \frac{\sum_i\,m_i v_{\ell,i}/(s_i\cos b_i)}{\sum_i
    m_i(R_0\cos\ell_i/(s_i\cos b_i)-1)},
\end{equation}
\begin{equation}
    \Omega_\mathrm{p}(b_n) = -\frac{\sum_i\,m_i v_{b,i}/s_i}{\sum_i
   m_i R_0 \sin\ell_i\sin b_i/s_i},
\end{equation}
\begin{equation}
    \Omega_\mathrm{p}(s_n) = \frac{\sum_i\,m_i v_{||,i}/s_i}{\sum_i
    m_iR_0 \sin\ell_i\cos b_i/s_i},
\end{equation}
where velocities are in the Galactic rest frame.
As in Section~\ref{Section::PatternSpeed}, we write the first of these expressions as
\begin{equation}
\begin{split}
       \Omega_\mathrm{p}(\ell_n) &= \frac{v_\odot-u_\odot\tan\ell_n+\mathcal{F}}{R_0-\mathcal{K}},\\\mathcal{F}&= \frac{\sum_i\,m_iv'_{\ell,i}/(s_i\cos b_i)}{\sum_i\,m_i\cos\ell_i/(s_i\cos b_i)},
       \>\mathcal{K}= \frac{\sum_i\,m_i}{m_i\cos\ell_i/(s_i\cos b_i)},
\end{split}
\end{equation}

We use the probabilistic model in equation~\eqref{Eqn::OmegaModel} to infer $|\mu_{\ell,A*}|$, $R_0$, $\Omega_\mathrm{p}$ and the distance systematic factor $\lambda$. The data covariance matrix $\boldsymbol{\Sigma}_{\mathcal{X}i}$ is computed using $100$ bootstrap resamples of the particle properties in each bin.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/perfect_sim.pdf}
    \caption{Inference of the pattern speed from simulation: corner plot shows the recovery of the pattern speed using two different priors on $R_0$ (blue from \protect\cite{McMillan2017} and red from \protect\cite{GravityCollab2018}). Above each panel we show the mean and standard deviation inferred for each parameter. The vertical/horizontal lines show the true values (in the top left panel we show the estimate of $\Omega_\mathrm{p}$ from the last $10$ snapshots). The inset shows the pattern speed estimator in bins of Galactic longitude for the true parameters. Horizontal lines correspond to those in the top left corner panel. $\lambda$ is a fractional distance systematic.}
    \label{fig:simulation_recovery}
\end{figure}

In Fig.~\ref{fig:simulation_recovery}, we show the inference for our model using the two Galactic centre distance priors. We have not applied any distance systematic to the simulation data, we use bins in $\ell$ of width $\Delta\ell=0.5\,\mathrm{deg}$ and we set $N_\mathrm{max}=2$. The inset shows the $\Omega_\mathrm{p}(\ell)$ estimate for each bin in $\ell$. We note that the central regions produce noisy estimates of $\Omega_\mathrm{p}$. This is probably because both numerator and denominator in the estimator are small, but could also be because the considered stars form part of the original bulge component which is perhaps not rotating with the bar. We also see that negative Galactic longitude produces more precise $\Omega_\mathrm{p}$ estimates as there are more stars in the solid angle considered. From the inference, we find that with both priors the pattern speed is well recovered (the expected pattern speed is $45\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$). The solar radius posterior follows the adopted prior, the proper motion of Sgr A* is tighter than the prior and the distance systematic is recovered as unbiased.

We apply our method to $9$ snapshots from the simulation where for each snapshot the bar is rotated to an angle $33\,\mathrm{deg}$ with respect to the line-of-sight. We then observe simulation particles within the VVV bulge region and use $N_\mathrm{max}=2$, $\Delta\ell=0.5\,\mathrm{deg}$ and $(\mu_{R0},\sigma_{R0})=(8.2,0.09)\,\mathrm{kpc}$. The results are shown in Fig.~\ref{fig:simulation_recovery_times}. We observe the decaying pattern speed of the bar. The recovery is shown with red errorbars (multiplied by $5$ for visibility). At all snapshots we recover the pattern speed with increasing precision at later times when the bar is more established. At early times, transient phenomena cause more uncertainty in the pattern speed. For $t<0.5\,\mathrm{Gyr}$ the pattern speed measured for all particles and those between $0.2$ and $0.5$ simulation units of the plane disagree slightly suggesting the bar hasn't reached equilibrium yet.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/sim_time_steps.pdf}
    \caption{Recovery of $\Omega_\mathrm{p}$ from our reference simulation at different times. The grey line shows the pattern speed inferred from the moments of inertia of consecutive snapshots. Red errorbars show the recovery from our modelling where we have multiplied the uncertainties by $5$ for visibility.}
    \label{fig:simulation_recovery_times}
\end{figure}

We perform further experiments varying (i) the systematic distance bias used to construct the mock data, (ii) the minimum $|\ell|$ considered, (iii) the maximum $|\ell|$ considered, (iv) the minimum $|b|$ considered, (v) the bin widths, (vi) number of polynomial terms in the model $N_\mathrm{max}$ and (vii) the width of the solar radius prior. The results are shown in Fig.~\ref{fig:simulation_recovery_parameters}. It is satisfying that varying most parameters does not bias the pattern speed significantly. Both distance systematics and uncertainty in the distance to the Galactic centre are irrelevant to the recovery (due to the degeneracy compensating one for the other). Using data with $|\ell|>\ell_\mathrm{min}$ or $|\ell|<\ell_\mathrm{max}$ does not alter the results other than increasing the uncertainty when less data is used (similar result for the bin size $\Delta\ell$) -- this is expected as the estimator $\Omega_\mathrm{p}(\ell)$ does not require coverage in $\ell$. The same is not true when considering only data with $|b|>b_\mathrm{max}$. The pattern speed is systematically biased when in-plane data is excluded. We previously checked the pattern speed of stars between $0.2$ and $0.5$ simulation units of the plane (approximately $b>3\,\mathrm{deg}$) was near identical to using all particles. It appears the recovery is satisfactory for $b_\mathrm{max}\lesssim1.5\,\mathrm{deg}$.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/sim_varying_parameters.pdf}
    \caption{Recovery of $\Omega_\mathrm{p}$ from our reference simulation using different setups: (i) including a relative distance systematic, (ii) using data with $|\ell|>\ell_\mathrm{min}$, (iii) using data with $|\ell|<\ell_\mathrm{max}$, (iv) using data with $|b|>b_\mathrm{min}$, (v) varying the bin width $\Delta\ell$, (vi) varying the number of polynomial coefficients in the model and (vii) increasing the width of the prior on $R_0$. The dark solid line is the `true' pattern speed and the fainter lines the pattern speed from the last $10$ snapshots.}
    \label{fig:simulation_recovery_parameters}
\end{figure}

We have also attempted to use the $\Omega_\mathrm{p}(b)$ estimator on the simulation but we find it is not reliable. This is possibly due to lack of resolution but also could be due to the boundary terms dominating the signal.

To summarise, we find that the recovery of $\Omega_\mathrm{p}$ is not affected by distance systematics, bin sizes, number of polynomial terms used $N_\mathrm{max}$, the range of $\ell$ used and the prior on $R_0$. If we filter out low latitude data, $|b|<b_\mathrm{min}$, we find the results are biased if $b_\mathrm{min}\gtrsim1.5\,\mathrm{deg}$. $R_0$ is completely degenerate with a distance systematic.

\subsection{Boundary terms}

When deriving the estimators for the pattern speed, we removed terms by arguing that they vanish at the boundaries. In realistic applications we are unable to integrate over all space so our estimators are biased by the non-zero contributions of the boundary terms. A boundary term near constant in $\ell$ will produce a constant offset in the Galactic centre distance or the motion of Sgr A*, whereas the unlikely case where boundary contributions behave like uncorrelated noise in $\ell$ will not bias the results due to our modelling excess scatter in $\mathcal{F}$ and $\mathcal{K}$. The concerning case is for near-linear variation in the boundary terms which masquerade as a change in the pattern speed.

When deriving the $\ell$-estimator, there are six boundary terms we discount:
\begin{equation}
    \begin{split}
       &(1)\>\>\>\>\>\>\>\>\Big[\int_{-\pi}^{\ell}\mathrm{d}\ell\int_{b_\mathrm{min}}^{b_\mathrm{max}}\mathrm{d}b\,s^2\rho v_{||}\cos b\Big]_{s_\mathrm{min}}^{s_\mathrm{max}},\\
       &(2)\>\>\>\>\>\>\>\>\Big[\int_{-\pi}^{\ell}\mathrm{d}\ell\int_{s_\mathrm{min}}^{s_\mathrm{max}}\mathrm{d}s\,s\rho v_{b}\cos b\Big]_{b_\mathrm{min}}^{b_\mathrm{max}},\\
       &(3)\>\>\>-\Big[\int_{-\pi}^{\ell}\mathrm{d}\ell\int_{b_\mathrm{min}}^{b_\mathrm{max}}\mathrm{d}b\,s^2\rho \cos^2b\sin\ell\Big]_{s_\mathrm{min}}^{s_\mathrm{max}}\Omega_\mathrm{p}R_0,\\
       &(4)\>\>\>\>\>\>\>\>\Big[\int_{-\pi}^{\ell}\mathrm{d}\ell\int_{s_\mathrm{min}}^{s_\mathrm{max}}\mathrm{d}s\,s\rho \sin b\cos b\sin\ell\Big]_{b_\mathrm{min}}^{b_\mathrm{max}}\Omega_\mathrm{p}R_0,\\
       &(5)\>\>\>-\Big[\int_{b_\mathrm{min}}^{b_\mathrm{max}}\int_{s_\mathrm{min}}^{s_\mathrm{max}}\mathrm{d}s\,s\rho(R_0\cos\ell-s\cos b)\Big]_{\ell=-\pi}\Omega_\mathrm{p},\\
       &(6)\>\>\>-\Big[\int_{b_\mathrm{min}}^{b_\mathrm{max}}\int_{s_\mathrm{min}}^{s_\mathrm{max}}\mathrm{d}s\,s\rho v_\ell\Big]_{\ell=-\pi}.
    \end{split}
\end{equation}
The first of these terms involves the line-of-sight velocities so in the absence of full spectroscopic coverage of the sky we must use simulations to estimate its amplitude. The second term involves the latitudinal velocities so in theory could be estimated from proper motion data, except we require proper motion data outside the observational volume to evaluate the $\ell$ integral.
The third and fourth terms involve no velocities so can be evaluated from the data modulo the same considerations about integrating over $\ell$. The final two terms are the lower limits of the $\ell$ integrals which cannot be evaluated from the data.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/boundary_terms_sim.pdf}
    \caption{Amplitude of the boundary terms neglected in our derivation of the pattern speed estimator $\Omega_\mathrm{p}(\ell)$. The thick black lines show the two terms used in the estimator ($\langle v_\ell \rangle_{b,s}$ [solid] and $\Omega_\mathrm{p}{\langle R_0\cos\ell-s\cos b\rangle_{b,s}}$ [dashed]), the grey line is their difference and the thinner black line is the sum of the neglected boundary terms.
    The neglected boundary terms are of order $10\,\mathrm{per cent}$ the terms used in the estimator.}
    \label{fig:boundary_terms}
\end{figure}

As we are unable to truly estimate the boundary terms from the data, we instead use our reference simulation. We convert the integrals into sums over particles as, for example,
\begin{equation}
\begin{split}
    &\Big[\int_{-\pi}^{\ell}\mathrm{d}\ell\int_{b_\mathrm{min}}^{b_\mathrm{max}}\mathrm{d}b\,s^2\rho v_{||}\cos b\Big]_{s_\mathrm{min}}^{s_\mathrm{max}}\\&\approx\frac{1}{2\Delta s}\Big[\int_{s-\Delta s}^{s+\Delta s}\mathrm{d}s'\,\int_{-\pi}^{\ell}\mathrm{d}\ell\int_{b_\mathrm{min}}^{b_\mathrm{max}}\mathrm{d}b\,s'^2\rho v_{||}\cos b\Big]_{s=s_\mathrm{min}}^{s=s_\mathrm{max}}\\&=\frac{1}{2\Delta s}\Big[\sum_i m_i\int_{s-\Delta s}^{s+\Delta s}\mathrm{d}s'\,\int_{-\pi}^{\ell}\mathrm{d}\ell\int_{b_\mathrm{min}}^{b_\mathrm{max}}\mathrm{d}b s'^2 v_{||}\cos b\delta(\boldsymbol{x}\!-\!\boldsymbol{x}_i)\Big]_{s=s_\mathrm{min}}^{s=s_\mathrm{max}}\\&=\frac{1}{2\Delta s}\Big[\sum_{i,(\ell_i,b_i,s_i)\in\mathcal{V}} m_i v_{i,||}\Big]_{s=s_\mathrm{min}}^{s=s_\mathrm{max}},
\end{split}
\end{equation}
where the sum is over the particles inside the volume $\mathcal{V}$ defined by $\ell=(-\pi,\ell), b=(b_\mathrm{min},b_\mathrm{max}),s'=(s-\Delta s,s+\Delta s)$. We set $\Delta s=1\,\mathrm{kpc}$ and $\Delta b=0.2\,\mathrm{deg}$.

In Fig.~\ref{fig:boundary_terms}, we show the amplitude of the boundary terms estimated from the reference simulation.
We use the pattern speed measured from consecutive snapshots.
We observe
the sum of the boundary terms is of order the difference in the estimator quantities and is approximately $10\,\mathrm{per cent}$ the magnitude of the estimator quantities. We see that near $\ell=0$ the boundary terms are significant relative to the estimator quantities as $v_\ell$ approaches zero here. This corresponds to the poor estimates of $\Omega_\mathrm{p}$ seen in Fig.~\ref{fig:simulation_recovery}. We have found that individual terms (1)-(4) in the sum of the boundary terms can be of order the estimator quantities but their sum is much smaller. When deriving the estimator formulae we assumed each of the terms was small but this does not appear to be true. It is perhaps fortuitous that their sum is negligible, but this appears to explain the degree of accuracy obtained through application of estimators to the simulation. The $b$-boundary terms can be made smaller if a symmetric interval is used, as we have done in the analysis of the data by assuming symmetry in $b=0$. Employing $b_\mathrm{max}=10\,\mathrm{deg}$ instead of $5\,\mathrm{deg}$ reduces the sum of the boundary terms to $\lesssim1\,\mathrm{per cent}$ for $\ell<0$ and $2-8\,\mathrm{per cent}$ for $\ell>0$. Additionally, we observe that the individual boundary terms, as well as their sum, are near constant with $\ell$ so will lead to systematic offsets in the properties of the Galactic centre rather than the pattern speed.

\section{Conclusions}\label{Section::Conclusions}

We have measured the pattern speed of the Milky Way bar as $\Omega_\mathrm{p}=(41\pm 3)\,\mathrm{km\,s^{-1}\,kpc^{-1}}$ using proper motion data from VVV and Gaia DR2.  This places corotation at
$(5.7\pm0.4)\,\mathrm{kpc}$. This result was obtained from the more reliable near-side of the bar and when the entire bar region is considered we obtain $\Omega_\mathrm{p}=(31\pm 1)\,\mathrm{km\,s^{-1}\,kpc^{-1}}$ but an inconsistent position and velocity of the Galactic centre. This suggests systematic uncertainties in our measurement of  $5-10\,\mathrm{km\,s^{-1}kpc^{-1}}$.

To establish this, we developed new estimators for the pattern speed using transverse velocity data derived from the Galactic proper motion components. These estimators use the Tremaine-Weinberg method of integrating the continuity equation. Our new estimators are tailored for use specifically in the Milky Way. Using our longitudinal velocity estimator, we build a probabilistic model that allows for full propagation of uncertainties. We have demonstrated the performance of the method through application to a disc galaxy simulation that has formed a dynamical bar. Although we only consider a selection of the simulation comparable to the VVV survey volume, we find our method robustly recovers the pattern speed at a number of simulation times. The only biases we detect are when excluding in-plane stars $|b|\lesssim1.5\,\mathrm{deg}$ when the method overestimates the pattern speed. When applying to data, only fields with $\ell>2.5\,\mathrm{deg}$ appear to produce reliable estimates possibly due to insufficient distance coverage, extinction effects or proper motion systematics.

\section*{Acknowledgements}
JLS thanks the Science and Technology Facilities Council, the Leverhulme Trust, the Newton Trust and Christ's College, Cambridge for financial support. We acknowledge useful conversations with HongSheng Zhao and Ortwin Gerhard, and we thank the anonymous referee for a close reading of the paper. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.

This paper made used of the Whole Sky Database (wsdb) created by Sergey Koposov and maintained at the Institute of Astronomy, Cambridge by Sergey Koposov, Vasily Belokurov and Wyn Evans with financial support from the Science \& Technology Facilities Council (STFC) and the European Research Council (ERC).

Based on data products from observations made with ESO Telescopes at the La Silla or Paranal Observatories under ESO programme ID 179.B-2002. This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the {\it Gaia} Multilateral Agreement. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the  Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science,
Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrof\'isica de Canarias, The Johns Hopkins University,
Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /
University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik Potsdam (AIP),
Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University,
New York University, University of Notre Dame,
Observat\'ario Nacional / MCTI, The Ohio State University,
Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group,
Universidad Nacional Aut\'onoma de M\'exico, University of Arizona,
University of Colorado Boulder, University of Oxford, University of Portsmouth,
University of Utah, University of Virginia, University of Washington, University of Wisconsin,
Vanderbilt University, and Yale University.

This publication made use of the Python science stack:
\texttt{numpy} \citep{numpy},
\texttt{scipy} \citep{scipy},
\texttt{matplotlib} \citep{matplotlib},
\texttt{ipython} \citep{ipython} and
\texttt{pandas} \citep{pandas}.

\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


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The Rib King
By Ladee Hubbard
Review by Thane Tierney
In the era of the belated (and semi-involuntary) retirement of the likes of Uncle Ben, Aunt Jemima and Mrs. Butterworth, The Rib King could hardly be more prescient, as it centers on a Black man who is the face of a food brand.
The novel's first half takes place near the beginning of World War I, a time when the Civil War was no further removed from memory than the Vietnam War is from our minds today. And while the formerly well-to-do white Barclay family is inclined to behave less spitefully toward people of different races, they are by no means paragons of enlightenment. Much as in the Depression-era classic My Man Godfrey, it turns out that the key to solving the family's financial ills may be held by the overlooked butler, in this case August Sitwell. He agrees to deliver a recipe for—and to be the public image of—a meat sauce that establishes him nationwide as the Rib King.
Fast forward a decade, and one of his former co-workers, Jennie Williams, has a product of her own to sell, which sweeps her unwillingly back into the Rib King's orbit. In this half of the book, Ladee Hubbard's talent really shines as Jennie navigates a maze of intrigue involving revenge, betrayal, economic exploitation, racial conflict and the often brutal exercise of power.
Hubbard's depiction of a shadow economy bracketed by race is compelling and insightful, reminiscent of playwright August Wilson's finest work. Woven into this narrative is a captivating depiction of Black feminist agency at a time not long after white women had gained the right to vote. It's little wonder that Hubbard won the Hurston/Wright Legacy Award for debut fiction in 2018.
Ultimately the reason to read The Rib King is not its timeliness or its insight into politics or Black culture, but because it accomplishes what the best fiction sets out to do: It drops you into a world you could not otherwise visit and makes you care deeply about what happens there.
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text:
Smart, BatStateU to power tsunami warning systems for seaside communities
PLDT's mobile unit Smart Communications is working with Batangas State University (BatStateU) to enhance tsunami warning systems of seaside communities. Smart is integrating its Emergency Cell Broadcast System (ECBS) into the Solar Isotropic Generator of Acoustic Wave (Sigaw) platform to enable faster and wider dissemination of hazard alerts within communities.
Funded by the World Food Programme, Sigaw was created by BatStateU, in partnership with the provincial disaster risk reduction and management office and Philippine Institute of Volcanology and Seismology (Phivolcs), to help warn residents in coastal barangays (villages) of possible tsunami.
Sigaw, which also means 'shout' in Tagalog, consists of a 10-foot-tall device, a control unit with a SIM and a siren, and a solar panel. It can send out text broadcasts and issue a siren alert when its sensors detect tsunami activity relayed by Phivolcs.
ECBS is a collaboration by Smart, the National Disaster Risk Reduction and Management Council, and the National Telecommunications Commission. It was launched in March 2017, and has since been used to disseminate quick emergency alerts during typhoons and post-earthquake situations. Most recently, ECBS alerts have been sent to Albay residents following the ongoing eruption of the Mayon Volcano, which started last month.
ECBS operates on a channel separate from that used by calls and text messages or SMS. 17 Sigaw units have been installed in 13 Batangas municipalities. The plan is to deploy more units not only in Batangas but also in other provinces.
Source: https://www.telecompaper.com/news/smart-batstateu-to-power-tsunami-warning-systems-for-seaside-communities–1233226
Batangas State University
ECBS
Emergency Cell Broadcast System
National Disaster Risk Reduction
PLDT's mobile
tsunami warning systems
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Thanks to the Star "Keep Alive" feature, the TSP134LAN can negotiate with any networked host and automatically print an error if the connection is lost due to computer/hub failure or cable damage. It also sends an immediate warning to the remote location that the printer is offline. "Real Time Status" alerts let know you when the printer's cover is open or when it runs out of paper. And because of the TSP134LAN's connectivity, any Mac or iPad on the network can see the printer's status.
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There's a new mural along Route 66 in Tulsa.
What does that mean? Well, that's a great question…and an important one in the overall story of Tulsa's success.
In 1901, oil was first discovered in what would become Tulsa County. When oil was struck at the Sue Bland #1 Well in the town of Red Fork, it rocketed 30 feet over the derrick in spectacular fashion. By the next week, thousands of men had poured into Indian Territory to seek their fortune. At that time only a railroad bridge connected Red Fork and Tulsa across the Arkansas River. The only way for most citizens to cross was by fording the river or using the ferry.
Tulsa city leaders recognized the need for a sturdy, reliable crossing across the Arkansas and tried to pass a bond issue to pay for a wagon bridge. The citizens of Tulsa, however, weren't convinced. The Red Fork oil strike hadn't resulted in a large discovery and many locals felt that the excitement was over.
Enter three local men: logger Melville Baird, merchant Joseph Don Hagler, and banker George T Williamson. They believed that oil was the future of Tulsa and pooled their own money to build a bridge across the Arkansas River. On January 4, 1904, the new wagon bridge opened to the public. On the top of the truss a small placard read, "You Said We Couldn't Do It, But We Did." It became a testament to Tulsa's resiliency and spirit.
In November of 1905, the Glenn Pool Oil Field was discovered. This monumental oil reserve is considered by many the true start to Oklahoma's oil boom and is what lead to Tulsa being billed as the Oil Capital of the World. If that wagon bridge had not been in place, it stands to reason that Sapulpa would instead be the second-largest city in Oklahoma and Tulsa would be the suburb.
In 1917, the wagon crossing was replaced with a state-of-the-art multi-span concrete arch bridge. It was a beautiful, modern construct with multiple lanes for traffic and rails in the middle to support the Inter-Urban Trolley. The new bridge was spearheaded by Tulsa County Commissioner Cyrus Avery, who was also heavily involved in the Good Roads Movement to improve lanes of travel throughout the country.
Avery's passion for roads lead to his appointment to the group tasked with designating and numbering the new federal highway system. This included Route 66, which Cyrus successfully lobbied to have routed through his home town of Tulsa. He used the 11th Street Bridge as a focal point for this decision, citing it as the best crossing across the Arkansas River in the country. The rest of the committee agreed and in 1926, the Mother Road was born.
You Said We Couldn't Do It, But We Did.
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Digital development resource: to buy or to build?
René Morency	 13/05/2019
Here's a statement to prompt a debate; in a volatile economy, it's more cost effective to have the pay-as-you-go services of a digital agency on call than it is to hire and build an in-house dev team.
After all, there's plenty of evidence that companies have cost savings on their mind right now, with many tightening their belts and putting a pause on spending. At the time of writing, investment by UK companies has fallen for three consecutive quarters. This is the first time this has happened, outside of a recession, in 15 years.
It's clearly a tough, nervous time for British businesses. The UK economy has had a turbulent and uncertain few years, and there is an increasing sense that these strains are coming to a head.
Consumers too seem to be cutting back. Just look at the High Street. First Poundworld shut its doors, followed by House of Fraser and Evans Cycles, while HMV, Patisserie Valerie and LK Bennett have all gone into administration since Christmas.
Hasn't it always been cheaper to outsource digital development?
Well, possibly, yes, but that doesn't always mean it's been the best thing to do.
Eighteen months ago, if you'd have pushed me, I'd probably have predicted the exact opposite to what I'm suggesting now; that we'd see a rise in businesses moving away from agencies and taking their digital services in-house.
At that time, the idea made sense. Digital products have become an integral part of remaining competitive for most organisations, and by investing in an in-house digital development team, companies can retaining knowledge within their business and potentially find new ways to apply digital products internally.
Building an in-house team also moves the cost of development services from being a capital expenditure to being an operational expenditure. This is a switch which can, for some organisations or teams, make it easier to obtain these services.
What's changed?
Well, while the benefits of bringing dev in-house are clear, so are the costs.
Consider that a digital team within a medium-sized business is probably a project manager, four engineers, a UX designer, a QA engineer, an analyst and a strategist. As you can imagine, the monthly salary, NI, pension and training costs for such a team add up quickly. Aside from the investment needed, adding a digital team to a company can also introduce challenges from an HR or cultural perspective.
In an increasingly volatile and uncertain business climate, these costs and challenges are starting to look like bad value for many companies.
To get an alternative perspective on the topic, I asked some of our customers this very question – why do you continue to use our services rather than build an in-house team? I was surprised by the unanimity of their answers.
Risk avoidance.
The argument is that in the current economy it's cheaper and far less risky to hire in specialists that can work on a project-by-project or retainer basis, allowing a company's capability to scale up or down rapidly.
For businesses that are sensitive to volatile changes in their markets, or for young companies that are trying to stay nimble, this way of working is a cost-effective, low-risk approach. Similar to the world of contractors, agencies – albeit in larger volume – can be turned on or off in line with demand.
While the calculation regarding whether to build an in-house digital team is complex and individual to each company, external macroeconomic factors are always going to influence such a decision, tilting the scales subtly one way or the other.
All other things being equal, the benefits of having an in-house team are clear, though for those with a risk-averse mindset, or those whose businesses are sensitive to fluctuations in global markets, the flexible, pay-as-you-go agency model is increasingly seen as not only an excellent short term solution but also a legitimate long-term business model.
About René Morency
MD of Browser London, René's core skills cover building, directing and managing teams, design insight, innovation and analysis and business strategy. He also has extensive experience creating, winning and retaining business opportunities.
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text:
Automation presents a political challenge, but also an opportunity
TechTank
Clara Hendrickson and William A. Galston Thursday, May 18, 2017
Donald Trump entered office on a promise to bring back jobs and lift up the white working-class. While 53 percent of Americans earning less than $30,000 a year voted for Hillary Clinton, the counties she won made up 64 percent of America's GDP in 2015. Trump, won over five times as many counties as Clinton, but these counties constituted only 36 percent of America's GDP in 2015.
Trump also surpassed Clinton in places where voters are most at risk of losing their jobs due to automation, and won support among those who had already lost out as a result of technological changes in the workplace. A National Bureau of Economic Research paper found that the more robots introduced in a commuting zone between 1993 and 2007, the more likely the area went to Trump.
Clara Hendrickson
Research Analyst - The Brookings Institution
Twitter clarajanehen
Ezra K. Zilkha Chair and Senior Fellow - Governance Studies
Twitter BillGalston
However, Trump's economic message and governance so far has focused almost entirely on job loss due to trade and he has said little on how automation challenges his constituency. On the other hand, Barack Obama warned in his final speech as president that "the next wave of economic dislocation won't come from overseas. It will come from the relentless pace of automation that makes many good, middle-class jobs obsolete." In the fall of 2016, the White House released a report on automation that outlined its likely impacts on lower-wage earners and the prospects for deepened wage inequality.
Research from Oxford University predicts 47 percent of jobs are likely to be automated within a decade or two, and a study by the McKinsey Global Institute finds that 46 percent of time spent on existing work activities will become automated. However, when asked about the threat of losing jobs to automation, Secretary Treasury Steve Mnuchin said the issue is "not even on our radar screen… I'm not worried at all." Not everyone agrees. Larry Summers, who originally argued that automation would help the economy by creating new jobs has since stepped back from this position, warning "This isn't some future possibility… This is something that's emerging before us right now." MIT economist Erik Brynjolfsson concurs: "This is the biggest challenge for our society for the next decade."
While much has been written on the policy challenges automation poses, little has been written on the political dynamics it is likely to produce. Unsurprisingly, political leaders tend to shy away from discussing the issue. It is difficult to antagonize technological advancement – automation is often the result of corporate investment, it yields growth and achieves efficiency, and average Americans save on products and services as a result. And some make the case that automation is not happening fast enough, citing dismal productivity gains.
The competing theses on the likely effects of automation differ in their understanding of how today's technological shifts compare to past radical changes in the economy. Nonetheless, policy recommendations that confront technological unemployment can insure against the worst-case scenario. And, as similar disruptions originate from trade and technological improvement, similar policies can ameliorate the consequences of dislocation produced by trade and tech shocks. Pursuing such policies could also present a political opportunity to restore some sober analysis and much-needed non-partisan problem-solving. This would require a response somewhat antithetical to our politics – one that is forward-thinking and anticipates voters' concerns.
Political solutions that anticipate change
While 65 percent of Americans believe that within 50 years, robots and computers will likely complete work performed by humans today, 80 percent of Americans do not expect their jobs will disappear. While poorer and less-educated Americans demonstrate greater concern for the prospect, American workers overall worry more about displacement by other workers, industry trends, and mismanagement. Given that few voters are preoccupied with disruptive changes to work stability through technological advancements, there exists little incentive for politicians to tackle it.
Will robots and AI take your job? The economic and political consequences of automation
Darrell M. West
How Millennials voted this election
William A. Galston and Clara Hendrickson
The state of self-driving car laws across the U.S.
Jack Karsten and Darrell West
But doing so provides two important opportunities at a time ripe for political renewal. First, addressing automation upends a politics that relies heavily on antagonism. Unlike trade, few voters reject the benefits of technological change, and automation offers no foreign scapegoats like cheap labor abroad or multilateral trade to accuse of stealing jobs. Subsequently, it is difficult to claim political victories and publically take on opponents. Second, while workers are concerned about more immediate threats, tackling automation would require modernizing the social safety net to not only improve the working lives of Americans today, but also to respond to the changing nature of work itself. As Brookings' Mark Muro writes, "programs should be anticipatory and offer support when disruption is forecast but not yet hitting home." This might require openness to challenging some of the premises that have long guided partisan economic platforms.
Because automation is predicted to yield greater economic growth while at the same time displacing many workers, traditional conservatives will have to embrace some degree of government intervention. An inability to adapt conservative economic governance to confront the features of radical shifts in the labor market could prove electorally costly. Americans likely to be the most affected by automation live in Republican districts, but in areas that have experienced the greatest shocks from automation, the 2016 presidential vote was closer. This presents an opportunity for Democrats to contend for the support of these voters. If Democrats fail to address these voters' needs, the party might miss an opportunity to reassert a strong economic message and reconnect with a working class base.
An opportunity and a cost
Joseph Parilla and Mark Muro have suggested the creation of a Universal Basic Adjustment Benefit to tackle the dislocation likely to become an increasingly routine feature of the labor market. Inspired by calls for a Universal Basic Income (UBI), the adjustment would include job-search counseling, worker training funding, relocation grants, and wage insurance. Given that UBI has attracted support from a group of odd bedfellows that includes American Enterprise Institute's Charles Murray and Andy Stern, the former president of the Service Employees International Union, a Basic Adjustment Benefit might be able to generate similar cross-ideological interest. In fact, it might even prove more palatable than UBI proposals as the benefit would target moving workers back into the labor market and would combine many welfare programs that could be administered flexibly at the local level.
The consequences of failing to incorporate automation into our political dialogue are potentially high. In the Midwest, manufacturing job loss bred feelings of economic frustration that led many to favor a populist candidate who gave them something to blame, which exemplifies the frustrations likely to take hold as the threat of technological unemployment mounts. Additionally, many of the jobs expected to disappear as a result of automation are held by white males without college educations, a group Trump won handily. Without efforts to ensure workers without a college degree can participate in the economy of the automated era, the growing education divide producing the kind of populist backlash here and across the west is likely to deepen.
Labor Policy & Unemployment


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