# Galactic Structure From Binary Pulsar Accelerations: Beyond Smooth Models

Thomas Donlon II  0000-0002-7746-8993 Department of Physics and Astronomy,
University of Alabama in Huntsville, 301 North Sparkman Drive, Huntsville, AL
35816, USA<EMAIL_ADDRESS>Sukanya Chakrabarti  0000-0001-6711-8140
Department of Physics and Astronomy, University of Alabama in Huntsville, 301
North Sparkman Drive, Huntsville, AL 35816, USA  Lawrence M. Widrow
0000-0001-6211-8635 Department of Physics, Engineering Physics and Astronomy,
Queen’s University, Kingston, ON K7L 3N6, Canada  Michael T. Lam
0000-0003-0721-651X SETI Institute, 339 N Bernardo Ave Suite 200, Mountain
View, CA 94043, USA  School of Physics and Astronomy, Rochester Institute of
Technology, Rochester, NY 14623, USA  Laboratory for Multiwavelength
Astrophysics, Rochester Institute of Technology, Rochester, NY 14623, USA
Philip Chang  0000-0002-2137-2837 Department of Physics, University of
Wisconsin-Milwaukee, 3135 N Maryland Ave, Milwaukee, WI 53211, USA  Alice C.
Quillen  0000-0003-1280-2054 Department of Physics and Astronomy, University
of Rochester, Rochester, NY 14627, USA

###### Abstract

We measure the line-of-sight accelerations of 26 binary pulsars due to the
Milky Way’s gravitational potential, and produce a 3-dimensional map of the
acceleration field of the Galaxy. Acceleration measurements directly give us
the change in the line-of-sight velocity at present day, without requiring any
assumptions inherent to kinematic modeling. We measure the Oort limit
($\rho_{0}=0.062\pm 0.017$ M⊙/pc3) and the dark matter density in the midplane
($\rho_{0,\textrm{DM}}=-0.010\pm 0.018$ M⊙/pc3); these values are similar to,
but have smaller uncertainties than previous pulsar timing measurements of
these quantities. Here, we provide for the first time, values for the Oort
constants and the slope of the rotation curve from direct acceleration
measurements. We find that $A=15.4\pm 2.6$ km/s/kpc and $B=-13.1\pm 2.6$
km/s/kpc (consistent with results from Gaia), and the slope of the rotation
curve near the Sun is $-2\pm 5$ km/s/kpc. We show that the Galactic
acceleration field is clearly asymmetric, but due to data limitations it is
not yet clear which physical processes drive this asymmetry. We provide
updated models of the Galactic potential that account for various sources of
disequilibrium; these models are incompatible with commonly used kinematic
potentials. This indicates that use of kinematically derived Galactic
potentials in precision tests (e.g., in tests of general relativity with
pulsar timing) may be subject to larger uncertainties than reported. The
acceleration data indicates that the mass of the Galaxy within the Solar
circle is $2.3\times 10^{11}$ M⊙, roughly twice as large as currently accepted
models. Additionally, the residuals of the acceleration data compared to
existing Galactic models have a dependence on radial position; this trend can
be explained if the Sun has an additional acceleration away from the Galactic
center.

##  I Introduction

The Milky Way (MW) is in dynamical disequilibrium due to internal phenomena,
such as the bar, spiral structure, and external perturbations from satellite
galaxies and dark matter substructure. The timescales for these perturbations
are comparable to the orbital periods of stars in the disk and stellar halo;
as a result, the Galaxy has not achieved relaxation in the statistical sense
described by Lynden-Bell (1967) [1]. These disequilibria manifest in the
stellar Galactic disk in a variety of ways, such as corrugations, warps, and
phase space spirals [2, 3, 4], as well as a north-south asymmetry in the
density of stars [5, 6, 7], and moving groups of disk stars [8, 9, 10, 11, 12,
13]. Disturbances in the HI gas [14, 15] have been interpreted as arising from
satellite perturbations [16, 17, 18], and there is apparent bulk motion in the
interstellar medium, such as the Radcliffe Wave [19].

These disequilibrium features contain valuable information about the MW’s
structure and history. For example, the phase space spirals in the MW disk may
have been generated by the Sagittarius Dwarf Galaxy [20] or the gravitational
field of the Galactic bar [21, 22]. In general, frequent interactions with
substructure leads to a departure from equilibrium in the motions of Milky Way
disk stars.

Historically, studies of the Galaxy’s structure have primarily focused on the
readily available positions and motions of stars [23]. This method is
restrictive, because kinematic information about the gravitational potential
is averaged over time and space throughout a given star’s orbit. Kinematic
techniques, such as Jeans modeling, require simplifying assumptions such as
spherical or azimuthal symmetry, and equilibrium [24]; these assumptions
restrict the ability to observe spatially complex and transient features, and
can therefore only produce an approximation of the Galaxy’s true underlying
gravitational potential field.

However, the acceleration of an object is instantaneous in time, and produces
a measure of the gravitational potential at a specific location of the Galaxy,
as well as a non-local measure of the density field. As a result, direct
acceleration measurements allow one to probe the Galaxy’s structure in a fine-
grained way that is free of kinematic assumptions. Recently, several precision
techniques have been developed that enable direct line-of-sight acceleration
measurements within a few kpc of the Sun, including extreme precision radial
velocity observations [25], pulsar timing [26], and eclipse timing [27].

Because they are precise astrophysical clocks, pulsars can now be used to
directly measure the Galactic acceleration [26]. The spin periods of solitary
pulsars are subject to magnetic braking, which is not well understood;
however, the orbital periods of binary pulsars can be modeled precisely with
general relativity [28, 29, 30] to extract Galactic accelerations.

Pulsar timing data has been used to test general relativity, via the emission
of gravitational waves from binary systems [29, 31]. Computing this radiation
term in the pulsar timing data requires knowing the Galactic potential at the
location of the pulsar, which is currently the largest source of uncertainty
in these tests [30]. By obtaining an accurate model of the Galaxy’s
gravitational potential, one could substantially lower the uncertainty in
tests of general relativity. This can be enabled by direct acceleration
measurements that are independent of pulsar timing. Eclipse timing
measurements of precisely timed eclipsing binary stars [27], which can be
employed for this purpose. Further in the future, extreme-precision radial
velocity measurements will also provide direct measurements of the Galactic
acceleration [25] that can be used to determine the Galactic potential
independent of pulsar timing.

Previously, Chakrabarti et al. (2021) [26, hereafter C21] analyzed line-of-
sight acceleration data for 14 binary pulsars; we expand this with updated
data for 26 binary pulsars, roughly doubling the size of the dataset. The C21
dataset spanned 2.6 kpc in $R$ and 1.5 kpc in $Z$; our updated dataset extends
this to 3.4 kpc in $R$ and 3.6 kpc in $Z$, which allows us to leverage a
larger region of the Galaxy when constraining our models.

In this work, we use the expanded dataset to compute updated values for
several different potential models that are fit to the observed accelerations.
In addition to a measurement of the Oort limit and a constraint on the local
dark matter density, we are now able to constrain the slope of the rotation
curve and the Oort Constants. We also move beyond smooth potential models, and
produce a 3-dimensional, non-parametric map of the MW acceleration field. This
map manifests an apparent lack of vertical and azimuthal reflection symmetry
about the Sun, which is evidence of local disequilibrium. We demonstrate that
a consideration of disequilibrium is required in order to obtain accurate
models of the MW’s gravitational potential from direct acceleration
measurements; this will continue to be important as pulsar timing datasets
become larger and more precise.

##  II Data

We take the distance between the Sun and the Galactic center to be
$R_{\odot}=8.178$ kpc [32], and the circular speed at the Solar position to be
$V_{LSR}=232.8$ km/s [33]. Adopting other literature values of $R_{\odot}$ and
$V_{LSR}$ did not substantially change the results of this work.

This work uses a Galactocentric Cartesian coordinate system, where the Sun is
located at positive $x$, $\hat{y}$ points towards the direction of the Sun’s
motion, and $\hat{z}=-\hat{x}\times\hat{y}$.

###  II.1 Calculating Accelerations

We follow the methodology in C21 in extracting the Galactic acceleration from
measured binary pulsar accelerations. The observed orbital period of a binary
pulsar will be Doppler shifted according to its line-of-sight velocity. If the
binary pulsar accelerates along our line of sight, a corresponding change in
the Doppler shifted orbital period will be observed. The observed line-of-
sight acceleration is expressed as

$a^{\textrm{Obs}}_{\textrm{los}}=\frac{\dot{P}_{b}^{\textrm{Obs}}}{P_{b}}c,$
(1)

where $P_{b}$ is the observed orbital period of the binary and $c$ is the
speed of light.

The observed time rate of change of the binary orbital period
($\dot{P}_{b}^{\textrm{Obs}}$) can be decomposed into a sum of different
effects:

$\dot{P}_{b}^{\textrm{Obs}}=\dot{P}_{b}^{\textrm{Shk}}+\dot{P}_{b}^{\textrm{GR}}+\dot{P}_{b}^{\textrm{Gal}}.$
(2)

The quantity $\dot{P}_{b}^{\textrm{Shk}}$ is known as the Shklovskii effect
[34], and is the result of proper motion of a source leading to a change in
the distance to that source, producing an apparent acceleration along our
line-of-sight. It is computed as

$\dot{P}_{b}^{\textrm{Shk}}=\frac{P_{b}\mu^{2}d}{c},$ (3)

where $\mu$ is the total proper motion of the source, and $d$ is the distance
to that source.

The quantity $\dot{P}_{b}^{\textrm{GR}}$ is the decrease in the orbital period
due to radiation of gravitational waves [28, 30], and is computed as

$\displaystyle\dot{P}_{b}^{\textrm{GR}}=-\frac{192\pi
G^{5/3}}{5c^{5}}\left(\frac{P_{b}}{2\pi}\right)^{-5/3}\left(1-e^{2}\right)^{-7/2}$
$\times\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right)\frac{m_{p}m_{c}}{\left(m_{p}+m_{c}\right)^{1/3}},$
(4)

where $e$ is the orbital eccentricity of the binary (typically close to 0),
$m_{p}$ is the mass of the pulsar, and $m_{c}$ is the mass of its companion.

Because the quantity we are interested in is the acceleration due to the
gravitational potential of the MW, we compute the Galactic component of the
line-of-sight acceleration as

$a^{\textrm{Gal}}_{\textrm{los}}=\frac{\dot{P}_{b}^{\textrm{Gal}}}{P_{b}}c$
(5)

using Equations 1-4. For notational simplicity, we often refer to this
quantity as $a_{\textrm{los}}$ from this point forwards.

###  II.2 Pulsar Data

Figure 1: The observed pulsar data. Each source is colored according to the
residual of its the observed line-of-sight acceleration and the Gala
MilkyWay2022 potential model, weighted by the uncertainty of each measurement.
There is a global trend in the data; points that are further to the left are
bluer on average than points that are further to the right. This suggests that
the Gala MilkyWay2022 potential model incorrectly estimates the slope of the
rotation curve, or that the Sun is experiencing an additional acceleration
away from the Galactic center.

We selected all available binary pulsars from the Australia Telescope National
Facility (ATNF) pulsar catalog [35] that satisfied the following criteria:

  * •

The source had a measured $P_{b}$ and $\dot{P}_{b}$;

  * •

Had measured parallax $\pi$ and proper motion $\mu$;

  * •

Had measured orbital eccentricity $e$, pulsar mass $m_{p}$, and companion mass
$m_{c}$; or a binary period $P_{b}>5$ days.

  * •

The source cannot be a redback, black widow, etc. or transferring mass with
its companion, which can change the orbital period of the system.

  * •

The source cannot be in a globular cluster, as the internal accelerations of
globular clusters are expected to overpower the Galactic acceleration signal.

PSR | Distance | $\dot{P}_{b}^{\textrm{Obs}}$ | $\dot{P}_{b}^{\textrm{Shk}}$ | $\dot{P}_{b}^{\textrm{GR}}$ | $a^{\textrm{Gal}}_{\textrm{los}}$ | Reference
---|---|---|---|---|---|---
| (kpc) |  |  |  | (mm s-1 yr-1) |
J0437-4715 | $0.157(2)$ | 3.728(6)E-12 | 3.76(5)E-12 | -3.22(15)E-16 | -0.8(9) | R16
J0613-0200 | $1.14(12)$ | 3.4(7)E-14 | 3.15(33)E-14 | -8.0(40)E-15 | 1.0(8) | NG
J0737-3039A/B | $1.15(22)$ | -1.252(17)E-12 | 4.4(12)E-16 | -1.2484(8)E-12 | -5.(18) | KMS06, DBT09
J0740+6620 | $1.0(2)$ | 1.17(26)E-12 | 1.06(21)E-12 | -6.07(21)E-16 | 2(8) | NG
J0751+1807 | $1.17(5)$ | -3.50(5)E-14 | 1.19(5)E-14 | -4.4(4)E-14 | -1.2(17) | EPTA, NSK08
J1012+5307 | $1.0(2)$ | 6.1(6)E-14 | 8.4(17)E-14 | -6.0(320)E-15 | -3(7) | NG, EPTA
J1017-7156 | $1.4(6)$ | 4.(2)E-13 | 2.0(8)E-13 | -3.9(30)E-16 | 3(4) | PPTA
J1022+1001 | $0.85(6)$ | 2.18(9)E-13 | 7.4(20)E-13 | -5.2(8)E-16 | -7.3(28) | EPTA
J1125-6014 | $1.5(11)$ | 7.(1)E-13 | 8.0(60)E-13 | -2.24(30)E-16 | -1(8) | PPTA
J1455-3330 | $0.76(6)$ | 4.5(22)E-12 | 8.0(6)E-13 | - | 5.3(32) | EPTA
B1534+12 | $0.94(7)$ | -1.366(3)E-13 | 5.3(4)E-14 | -1.93(6)E-13 | 0.8(19) | FST14, DDF21
J1600-3053 | $1.87(3)$ | 5.(1)E-13 | 2.79(5)E-13 | -1.2(4)E-16 | 1.7(8) | PPTA
J1603-7202 | $2.7(13)$ | 1.9(4)E-13 | 2.2(10)E-13 | - | -0.4(19) | PPTA, WRT22
J1614-2230 | $0.68(4)$ | 1.33(7)E-12 | 1.32(8)E-12 | -4.23(3)E-16 | 0.1(14) | NG
J1640+2224 | $1.08(28)$ | 9.5(19)E-12 | 5.3(14)E-12 | -6.(3)E-18 | 2.6(15) | EPTA
J1713+0747 | $1.136(13)$ | 2.64(73)E-13 | 6.41(7)E-13 | -6.7(2)E-18 | -0.61(12) | EPTA
J1738+0333 | $1.47(10)$ | -1.70(31)E-14 | 8.2(6)E-15 | -2.9(13)E-14 | 1(4) | FWE12
J1741+1351 | $3.0(9)$ | 1.3(4)E-12 | 1.4(4)E-12 | - | -1(4) | NG
J1909-3744 | $1.12(3)$ | 5.09(2)E-13 | 4.94(13)E-13 | -2.90(4)E-15 | 1.2(10) | NG
B1913+16 | $4.1(2)$ | -2.423(1)E-12 | 1.6(7)E-16 | -2.40263(5)E-12 | -6.96(34) | WH16, DWN18
J1933-6211 | $1.6(3)$ | 7.(1)E-13 | 6.6(12)E-13 | -1.53(29)E-16 | 0.3(14) | GKF23
J2043+1711 | $1.4(1)$ | 1.0(1)E-13 | 6.5(5)E-14 | -2.86(16)E-15 | 2.8(8) | NG
J2129-5721 | $3.6(14)$ | 1.51(9)E-12 | 8.9(35)E-13 | - | 10(6) | PPTA
J2145-0750 | $0.71(5)$ | 1.3(2)E-13 | 1.76(12)E-13 | - | -0.7(4) | PPTA
J2222-0137 | $0.2681(12)$ | 2.554(74)E-13 | 2.796(13)E-13 | -7.98(12)E-15 | -0.76(34) | GFG21
J2234+0611 | $0.97(4)$ | 3.1(25)E-12 | 4.78(20)E-12 | -2.57(12)E-17 | -6(9) | SFA19

Rows without $\dot{P}_{b}^{\textrm{GR}}$ data are expected to have negligibly
small values of $\dot{P}_{b}^{\textrm{GR}}$. References: R16, [36]; NG
(NANOGrav 15-yr Data Release), [37]; KMS06, [38]; DBT09, [39]; EPTA (EPTA-
DR2), [40]; NSK08, [41]; FST14, [42]; DDF21, [43]; PPTA (PPTA-DR2), [44];
WRT22, [45]; FWE12, [46]; WH16, [30]; DWN18, [47]; GKF23, [48]; GFG21, [49];
SFA19, [50].

Table 1: Summary of Pulsar Data.

Our final dataset contains 26 binary pulsars; the full list of the pulsars,
their distances, their components of $\dot{P}$, and their accelerations are
provided in Table 1. Fourteen of these sources are present in the C21 dataset.
Here, we also include 9 sources from recent work by Moran  _et al._ 51. We do
not use the following five sources from the Moran et al. (2023) dataset as
they do not satisfy our criteria for selection:

  * •

B1259-63: This pulsar has a Be star companion, and experiences regular mass
transfer [52].

  * •

J0348+0432: This source has no PTA parallax available. Moran  _et al._ 51 used
a Bayesian estimate of the parallax taken from a negative Gaia DR3 parallax
[53, 54, 55] to calculate the distance to the pulsar, but we choose not to
include this source because of its large distance uncertainty.

  * •

J0636+5128 (Listed in Moran  _et al._ 51 as J0636+5129): There is evidence for
outflows around this source, the pulsar may have an accretion disk, and the
pulsar appears to be interacting with its companion in some way [56, 57].

  * •

J1756-2251: The proper motion in declination of this source is only
constrained within an upper limit of 20 mas; Moran  _et al._ 51 use only the
proper motion in right ascension for this source (roughly 2 mas).
Additionally, the distance to this source has an uncertainty of 24 kpc. This
makes it impossible to constrain $\dot{P}_{b}^{\textrm{Shk}}$.

  * •

J2339-0533: This source is a redback [58].

Three of these pulsars are new sources, and were not in either the C21 or
Moran  _et al._ 51 datasets: J1455-3330, J1640+2224, and J1933-6211; these
timing solutions were published after Moran  _et al._ 51.

C21 listed a parallax of 1.9 mas for J2129-5721 [59], but more recent
solutions give a parallax of 0.26 mas for this source [44]. This erroneous
distance measurement, combined with the source’s large value of
$\dot{P}^{\textrm{Obs}}_{b}$, could explain why it had the largest residual of
any PSR considered by C21.

In the case where $e$, $m_{p}$ and $m_{c}$ were not constrained (and therefore
$\dot{P}_{b}^{\textrm{GR}}$ could not be calculated), the source was still
included if its binary period $P_{b}>5$ days. Since
$\dot{P}_{b}^{\textrm{GR}}\propto P_{b}^{-5/3}$, large orbital periods result
in negligibly small values of $\dot{P}_{b}^{\textrm{GR}}$ (assuming $m_{p}$ =
$m_{c}$ = $1.4$ M⊙, $e=0$, and $P_{b}=5$ days, $\dot{P}_{b}^{\textrm{GR}}\sim
10^{-15}$; for comparison, $\dot{P}_{b}^{\textrm{Gal}}\sim 10^{-12}$). Our
dataset contains 5 sources without $\dot{P}_{b}^{\textrm{GR}}$, indicated in
Table 1 by a dash in the $\dot{P}_{b}^{\textrm{GR}}$ column.

###  II.3 Line-of-Sight Acceleration Uncertainty

Figure 2: Bootstrapped uncertainties in $a_{\textrm{los}}$ for each source,
broken down by individual component (parallax, proper motion,
$\dot{P}^{\textrm{GR}}_{b}$, and $\dot{P}^{\textrm{Obs}}_{b}$). There is also
a contribution to $a_{\textrm{los}}$ error from covariances between the
various components. The dashed black line shows the uncertainty estimated for
$a_{\textrm{los}}$ using error propagation techniques, and is generally in
agreement with the bootstrapped uncertainties.

We computed the uncertainty in $a_{\textrm{los}}$ for each source using
standard error propagation procedure. This method does not take into account
any covariances between the uncertainties in correlated parameters, or the
fact that uncertainty in distance contibutes to the total error by both
changing the position of the pulsar and also affecting the calculation of
$\dot{P}_{b}^{\textrm{Shk}}$. As such, it is beneficial to double check that
we are not underestimating the uncertanties in $a_{\textrm{los}}$ by
estimating the uncertainties in a second way.

Figure 2 shows bootstrapped uncertainties in $a_{\textrm{los}}$ for each
source, broken down into the contribution of the uncertainty of each component
to the total $a_{\textrm{los}}$ uncertainty. The total uncertainty in
$a_{\textrm{los}}$ was estimated by computing the “true” $a_{\textrm{los}}$
for each source assuming its parameters each individually have zero error, and
then sampling 10,000 times with the reported errors on each component,
assuming that each uncertainty is normally distributed. The contribution of
each component to the overall $a_{\textrm{los}}$ uncertainty was estimated by
assuming only that component has zero uncertainty, and then sampling the
resulting $a_{\textrm{los}}$ uncertainty 10,000 times. The “Other” component
in the $a_{\textrm{los}}$ uncertainties is likely due to additional
uncertainty contributed by covariances in the components.

The dashed black line in Figure 2 shows the uncertainty that is obtained
through error propagation; overall this value appears to be similar to the
bootstrap estimates, indicating that the uncertainties obtained via error
propagation are reasonable estimates.

In general, uncertainties in parallax and $\dot{P}^{\textrm{Obs}}_{b}$
contribute more to the overall uncertainty in $a_{\textrm{los}}$ than
uncertainties in proper motion and $\dot{P}^{\textrm{GR}}_{b}$. As a result,
improvements in parallax and $\dot{P}^{\textrm{Obs}}_{b}$ measurements should
be prioritized in pulsar timing fits in order to obtain more precise
$a_{\textrm{los}}$ measurements in the future.

##  III Radial Dependence of Acceleration Residuals

Figure 3: The residuals of the pulsar data and the Gala MilkyWayPotential2022
model as a function of distance from the Galactic center (black). A linear fit
to the data is shown as a dashed gray line. If the model fits the data, one
would expect the residuals to be clustered around 0 independent of each
pulsar’s position; however, there is a radial dependence in the residuals when
the Solar acceleration from the Gala model is used. If the Sun’s acceleration
relative to the binary pulsars is actually only 80% of its presumed value,
this trend disappears (although there is still substantial scatter at some
radii, particularly at $R$ $\sim$ 7.5 kpc).

The observed acceleration data is shown in Figure 1 compared to a state-of-
the-art theoretical potential model, MilkyWayPotential2022 from the `Gala`
python package [60]. There is a global trend in the figure; overall,
MilkyWayPotential2022 underestimates the accelerations of the pulsar sources
towards the Galactic center. Because we observe a line-of-sight acceleration
relative to the Sun, this results in a positive residual $a_{\textrm{los}}$
(blue) for sources between the Sun and the Galactic center, and a negative
residual $a_{\textrm{los}}$ (red) for sources with $R>R_{\odot}$.

Interestingly, we were not able to remove this trend by varying the total mass
of the Gala MilkyWayPotential2022 model. However, this trend can be explained
if the Sun is accelerating relative to the pulsars in the dataset.

The distribution of QSOs has been used to measure the acceleration of the Sun
relative to a cosmological reference frame [61, 62]. These accelerations agree
with predictions of the Solar acceleration relative to the Galactic center
using existing kinematic potential models of the MW; as a result, the measured
acceleration of the Sun relative to the QSO reference frame is typically
treated as identical to the Sun’s acceleration relative to the Galactic
center. In other words, it is presumed that the acceleration of the Galaxy
relative to the QSO reference frame is negligible. However, there is no reason
that the Galactic center is in the same inertial reference frame as the QSOs;
if the Galactic center is in-fact accelerating relative to the QSOs, then the
acceleration of the Sun relative to the QSOs is not a relative acceleration
with respect to the Galactic center.

In this scenario, the Sun could be experiencing an additional acceleration
relative to the Galactic center (which cannot be constrained by the QSO
measurements). If this is the case, then the observed relative accelerations
of each pulsar are actually

$a_{\textrm{los}}=\left[\mathbf{a}(\mathbf{x})-\mathbf{a}(\mathbf{x}_{\odot})-\mathbf{a}_{\textrm{pec}}(\mathbf{x}_{\odot})\right]\cdot\hat{\mathbf{d}},$
(6)

where $\mathbf{a}_{\textrm{pec}}(\mathbf{x}_{\odot})$ is a peculiar
acceleration felt by the Sun but not felt by each pulsar (see Equation 8 for
further context) and $\hat{\mathbf{d}}$ is the line-of-sight vector from the
Sun to each source.

In order to remove the observed trend, the Sun would need to feel an
additional acceleration of $\mathbf{a}_{\textrm{pec}}(\mathbf{x}_{\odot})=1.1$
mm/s/yr of its predicted radial acceleration away from the center of the
Galaxy. The effect of this additional acceleration on the residual of the
observed data is shown in Figure 3. This value was obtained by optimizing a
potential model that consisted of MilkyWayPotential2022 plus an additional
acceleration on the Sun relative to the pulsar data (see Section IV for a
detailed description of the optimization procedure). For this analysis, we
removed PSRs J1713+0747, B1913+16, and J2043+1711 because they were outliers
that substantially changed the linear fit to the data.

It is difficult to explain why the Sun would have an additional acceleration
relative to the Galactic center, although a relative acceleration could
potentially be caused by a nearby massive object, such as a nearby (dim)
compact object, or a dark matter subhalo. It is interesting to note that a
Jupiter-mass object at a distance of 400 AU from the Sun would produce a Solar
acceleration of the correct magnitude to explain the observed global pulsar
acceleration trend; constraints on the possible locations of Planet Nine place
it roughly in the direction of the Galactic anticenter [63, 64], potentially
making it an explanation for why the Sun would experience an additional
acceleration away from the center of the Galaxy that would not be felt by the
pulsar sources. However, current constraints on Planet Nine’s mass and
semimajor axis would only produce a Solar acceleration on the order of
$\sim$0.05 mm/s/yr [65]; this discrepancy would need to be resolved in order
to claim that a ninth planet is the cause of the apparent 1.1 mm/s/yr Solar
acceleration.

Zakamska & Tremaine (2005) [66] used pulsar and white dwarf timing solutions
to constrain the acceleration of the Solar system, and concluded that the
Solar system acceleration was consistent with zero. However, their sensitivity
was limited to roughly the acceleration of a Jupiter-mass planet at a distance
of 200 AU ($\sim 4.5$ mm/s/yr), which is a few times larger than the value we
provide here for the Solar acceleration, and is therefore not in tension with
our results.

Local Model | Potential | Parameters | AIC | $\chi^{2}$
---|---|---|---|---
Equilibrium Models |  |  |
$\alpha$ | $\Phi(R,z)=V^{2}_{LSR}\ln\left(\frac{R}{R_{\odot}}\right)+\frac{1}{2}\alpha z^{2}$ | $\log_{10}(\alpha\cdot\textrm{Gyr}^{2})=3.60\pm 0.06$ | 54 | 2.2
$\alpha$-$\beta$ | $\Phi(R,z)=\frac{V^{2}_{LSR}}{2\beta}\left(\frac{R}{R_{\odot}}\right)^{2\beta}+\frac{1}{2}\alpha z^{2}$ | $\log_{10}(\alpha\cdot\textrm{Gyr}^{2})=3.51\pm 0.13$ | 55 | 2.3
|  | $\beta=0.14\pm 0.17$ |  |
Cross ($\alpha$-$\gamma$) | $\Phi(R,z)=\left(V^{2}_{LSR}-\gamma z^{2}\right)\ln\left(\frac{R}{R_{\odot}}\right)+\frac{1}{2}\alpha z^{2}$ | $\log_{10}(\alpha\cdot\textrm{Gyr}^{2})=3.54\pm 0.16$ | 55 | 2.3
|  | $\log_{10}(\gamma\cdot\textrm{Gyr}^{2})=3.3\pm 0.9$ |  |
MWPotential2014 |  Galpy Values [67] | - | 66 | 3.0
MilkyWayPotential2022 |  Gala Values [60] | - | 70 | 3.5
Damour-Taylor | see Section IV.1.4 | - | 85 | 3.8
Disequilibrium Models |  |  |
Anharmonic | $\Phi(R,z)=V^{2}_{LSR}\ln\left(\frac{R}{R_{\odot}}\right)+\frac{1}{2}\alpha_{1}z^{2}+\frac{1}{3}\alpha_{2}z^{3}$ | $\log_{10}(\alpha_{1}\cdot\textrm{Gyr}^{2})=3.51\pm 0.08$ | 51 | 2.1
|  | $\log_{10}(\alpha_{2}\cdot\textrm{Gyr}^{2}\cdot\textrm{kpc})=3.4\pm 0.2$ |  |
$2\alpha$-$\beta$ | $\Phi(R,z)=\frac{V^{2}_{LSR}}{2\beta}\left(\frac{R}{R_{\odot}}\right)^{2\beta}+\frac{1}{2}\alpha_{1}z^{2}+\frac{1}{3}\alpha_{2}z^{3}$ | $\log_{10}(\alpha_{1}\cdot\textrm{Gyr}^{2})=3.56\pm 0.11$ | 53 | 2.2
|  | $\log_{10}(\alpha_{2}\cdot\textrm{Gyr}^{2}\cdot\textrm{kpc})=3.5\pm 0.2$ |  |
|  | $\beta=-0.08\pm 0.18$ |  |
Local Expansion | see Section V.1.2 | $\log_{10}(-\partial a_{R}/\partial R\cdot\textrm{Gyr}^{2}/\textrm{kpc})=2.6\pm 0.3$ | 41 | 1.6
|  | $\log_{10}(-\partial a_{\phi}/\partial\phi\cdot\textrm{Gyr}^{2}/\textrm{kpc})=3.54\pm 0.09$ |  |
|  | $\log_{10}(\partial a_{z}/\partial z\cdot\textrm{Gyr}^{2}/\textrm{kpc})=3.73\pm 0.08$ |  |
Sinusoidal | $\Phi(R,z)=V^{2}_{LSR}\ln\left(\frac{R}{R_{\odot}}\right)$ | $\log_{10}(\alpha\cdot\textrm{Gyr}^{2})=3.37\pm 0.08$ | 41 | 1.6
|  $+\frac{1}{2}\alpha\left(z+A\sin\left(2\pi R/\lambda+\varphi\right)\right)^{2}$ |  $A=0.52\pm 0.08$ kpc |  |
|  |  $\lambda=2.46\pm 0.15$ kpc |  |
|  | $\varphi=3.5\pm 1.2$ |  |
$\alpha$-$\beta$ \+ 2 Point Mass | $\Phi(R,z)=\frac{V^{2}_{LSR}}{2\beta}\left(\frac{R}{R_{\odot}}\right)^{2\beta}+\frac{1}{2}\alpha z^{2}+\sum_{i}^{2}\frac{GM_{i}}{|\vec{r}-\vec{r_{i}}|}$ | $\log_{10}(\alpha\cdot\textrm{Gyr}^{2})=3.65\pm 0.09$ | 31 | 0.8
|  | $\beta=-0.43\pm 0.12$ |  |
|  | $\log_{10}(M_{1}/M_{\odot})=8.3\pm 0.3$ |  |
|  |  $x_{1}=7.4\pm 0.2$ kpc |  |
|  |  $y_{1}=1.2\pm 0.1$ kpc |  |
|  |  $z_{1}=-0.6\pm 0.2$ kpc |  |
|  | $\log_{10}(M_{2}/M_{\odot})=8.0\pm 0.5$ |  |
|  |  $x_{2}=7.5\pm 0.1$ kpc |  |
|  |  $y_{2}=0.5\pm 0.1$ kpc |  |
|  |  $z_{2}=0.9\pm 0.2$ kpc |  |
Table 2: Optimized fits to the local potential models.

##  IV Model Fits

Ideally, if one had a complete map of the gravitational field, then its
divergence would give a local measure of the density field; however, because
we are restricted to one component of the gravitational field with existing
data, we must instead rely on models to infer the local density and
corresponding features in the Milky Way.

Given a model for the MW’s gravitational potential $\Phi(\vec{x})$, one can
compute the acceleration at a given point from

$\mathbf{a}(\mathbf{x})=-\nabla\Phi(\mathbf{x}),$ (7)

which can then be used to obtain a relative line-of-sight acceleration:

$a_{\textrm{los}}=\left[\mathbf{a}(\mathbf{x})-\mathbf{a}(\mathbf{x}_{\odot})\right]\cdot\hat{\mathbf{d}}$
(8)

We fit a number of models to the observed $a_{\textrm{los}}$ data using a
maximum likelihood estimation (MLE) technique [68]. We first define a
likelihood function, in this case obtained from a $\chi^{2}$ statistic:

$\chi^{2}=\sum_{i}^{N}\frac{\left(a_{\textrm{los}}^{\textrm{obs}}-a_{\textrm{los}}^{\textrm{model}}\right)^{2}}{\sigma_{a_{\textrm{los}}^{\textrm{obs}}}^{2}},$
(9)

where $N$ is the number of sources in the data. The corresponding log-
likelihood function is $\ln(L)=-\chi^{2}/2.$

The log-likelihood function was then maximized using the
`differential_evolution` function from the `scipy.optimize` python package
[69]. Differential evolution was used rather than traditional conjugate
gradient descent because some of the models have “bumpy” likelihood surfaces,
which can cause gradient descent algorithms to get stuck in local minima.

The covariance matrix ($K$) for the optimized parameters is calculated from
the weighted inverse of the Hessian matrix ($H$);

$K_{ij}=\frac{1}{N}H_{ij}^{-1},$ (10)

where $N$ is again the number of sources, and

$H_{ij}=\left[\frac{\partial^{2}}{\partial_{i}\partial_{j}}\ln(\hat{L})\right],$
(11)

where $\hat{L}$ is the likelihood of the optimized parameters given the
observed data. The uncertainty in the $i$th optimized parameter ($\sigma_{i}$)
is obtained from the diagonal of the covariance matrix:

$\sigma_{i}=\textrm{diag}(K_{ij}).$ (12)

The Akaike Information Criterion (AIC) can then be evaluated for each fit
model using the formula

$\textrm{AIC}=2k-2\ln(\hat{L})=2k+\chi^{2},$ (13)

where $k$ is the number of free parameters in that model. The AIC is a measure
of significance of a likelihood score given a model; models with more free
parameters typically produce better likelihood scores, so the AIC provides a
way of evaluating whether adding additional free parameters to a model
actually improves the quality of the fit. A difference of $\Delta$AIC$=6$
between two models is considered to be strong evidence in favor of the model
with lower AIC [70].

C21 used a Markov Chain Monte Carlo (MCMC) technique to fit models to pulsar
acceleration data. We chose to use an MLE technique instead because MCMC codes
can quickly become intractable for models with a large number of parameters,
such as the “$\alpha-\beta$ \+ 2 Point Mass” model, which has 10 free
parameters. This model is feasible in the MLE code, which is much faster and
uses much less memory than the corresponding MCMC framework. The C21 MCMC
setup is practically identical to our MLE setup, in that the likelihood
functions are identical, and the MCMC priors from C21 are (roughly) encoded in
the uncertainties in the observed accelerations, which are derived from the
usual uncertainty propagation procedure. We compared several of our model fits
to fits obtained using C21’s MCMC approach, and the optimized parameters agree
to well within the reported uncertainties.

The analytic form of each potential model can be found in Tables 2 and 3, as
well as the best fit parameters for each model and the quality of that fit.

###  IV.1 Symmetric Local Models

Our first set of potential models are azimuthally and vertically symmetric
models (they are invariant under negation of $\phi$ or $z$). These models were
introduced in C21 as local approximations of the potential (Taylor expansions
of order $\leq 2$), which are only expected to be accurate within a few kpc of
the Sun. For this reason, these models are only fit to sources within 3 kpc of
the Sun; this removes J2129-5721 and B1913+16 from the dataset, and these
models are only fit to 24 of the 26 sources.

####  IV.1.1 $\alpha$ Model

The $\alpha$ model is the simplest model in this work. It is additively
separable in $R$ and $z$, and assumes a flat rotation curve, which is
calibrated to the local standard of rest and the Solar position. In the
vertical direction, the potential is that of a harmonic oscillator, where
$\alpha$ gives the angular frequency of vertical oscillations. The free
parameter $\alpha$ also sets the vertical density profile of this model, which
is constant and given by $4\pi G\rho=\alpha$. Our measured value of $\alpha$
agrees with the value obtained by C21, and our updated value has smaller
uncertainties.

####  IV.1.2 $\alpha-\beta$ Model

The $\alpha-\beta$ model is an extension of the $\alpha$ model that considers
a sloped rotation curve, which is characterized by $\beta$:

$\beta=\left[\frac{R}{V_{\textrm{circ}}}\derivative{V_{\textrm{circ}}}{R}\right]_{R_{\odot}},$
(14)

i.e. more positive (negative) values of $\beta$ indicate a more positive
(negative) slope of the rotation curve. Given the smaller extent of the
pulsars, C21 could not constrain $\beta$ for this model. Our larger of sample
of pulsars covering a larger radial extent enables us to constrain it to be
slightly positive (although it is consistent with 0 within a 1$\sigma$
uncertainty). Constraining $\beta$ allows us to calculate the Oort constants;
we provide an updated estimate of the Oort limit as well (see Section VI).

####  IV.1.3 “Cross” Model

A potential that is additively separable in $R$ and $z$ (such as the $\alpha$
model) leads to a density that is also additively separable through the
Poisson equation, whereas any realistic potential will not satisfy this
condition. In the “cross” model, we consider a simple extension of the
$\alpha$ model that breaks the separability assumption. The magnitude of this
cross term is set by $\gamma$, for which $\gamma>0$ corresponds to oblate
isopotential contours, and a larger value of $\gamma$ implies a more oblate
potential.

C21 derive typical values of $\gamma$ to be
$\log_{10}(\gamma/\textrm{Gyr}^{2})=2.93$ for a log-spherical MW potential and
$\log_{10}(\gamma/\textrm{Gyr}^{2})=3.94$ for a Miyamoto-Nagai Disk potential
[71]. Both of these values are consistent with our fit value of $\gamma$,
although our observed value of $\gamma$ is closer to the spheroid potential
than the disk potential. C21 found that their value of $\gamma$ was more disky
($\log_{10}(\gamma)=4.87$) than our best-fit $\gamma$ value
($\log_{10}(\gamma)=3.3$). The difference in these values arises from the
updated timing solutions for the pulsars relative to what was used in C21;
when fitting the cross model to only the C21 sources, but instead using the
more recent values for the accelerations and distances, we obtain a $\gamma$
value that is consistent with our $\gamma$ within 1$\sigma$ uncertainties.

Figure 4: The ratio of $V_{\textrm{circ}}$ to $V_{\textrm{LSR}}$ as a function
of height from the disk. The dashed red (this work) and blue (C21) lines show
the values obtained from the cross model using pulsar acceleration data, and
the solid black line shows the values obtained from kinematic data by Sharma
et al. (2014) [72]. The vertical extent of the pulsar data (which is roughly
equal for C21 and our local dataset) is shown as a vertical dotted line. The
C21 pulsar data produces a more disky model than our updated model which uses
more recent timing solutions, and which is now consistent with the kinematic
values from Sharma  _et al._ 72 to the vertical extent of the data.

The cross model also gives an approximation for the circular speed as a
function of $Z$ at $R=R_{\odot}$:

$\frac{V_{\textrm{circ}}}{V_{\textrm{LSR}}}\approx
1-\frac{|\gamma|}{2V^{2}_{\textrm{LSR}}}z^{2}.$ (15)

This expression is plotted in Figure 4 for our best-fit value of $\gamma$,
along with the best-fit $\gamma$ value from C21, and a model of
$V_{\textrm{circ}}/V_{\textrm{LSR}}$ from Sharma  _et al._ 72, who analyzed
kinematic data of MW stars. The Sharma  _et al._ 72 expression closely agrees
with canonical MW potential models, such as the model of Law & Majewski (2010)
[73]. Overall, $V_{\textrm{circ}}/V_{\textrm{LSR}}$ is similar for our pulsar
data and the kinematic data; however, the C21 $\gamma$ value produces a very
disky potential model that is inconsistent with the kinematic data from Sharma
_et al._ 72. As our local dataset has the same vertical extent as the C21
dataset, we conclude that these differences are due to improvements in the
reported pulsar timing values.

It is possible to relate $\gamma$ and the scale length of a Miyamoto-Nagai
Disk potential with Eq. 15 of C21, by fixing the mass and scale length of the
disk to the C21 values, and then numerically solving for the scale height
given some value of $\gamma$. This corresponds to a scale height of 1.0 kpc
for our fit $\gamma$ value, which is somewhat larger than but generally
consistent with MW thick disk populations. The C21 $\gamma$ measurement
corresponds to a scale height of 0.05 kpc, implying a much more condensed disk
than we find using the updated pulsar dataset.

####  IV.1.4 Damour-Taylor Model

The orbital decay via the emission of gravitational radiation by binary
pulsars can be used to test general relativity. The Damour-Taylor model [29]
is an approximation to the MW potential that is commonly used in these studies
[29, 74, 75]. While its functional form has changed somewhat since its
original definition in Damour and Taylor 29, the model is essentially a flat
rotation curve model with a roughly linear disk acceleration profile, which is
similar to our $\alpha$ model.

We use the version of the model from Lazaridis  _et al._ 75, where the
acceleration profile is defined as:

$\displaystyle a_{\textrm{los}}=$
$\displaystyle\;\;\;\;\;a_{R,\textrm{los}}+a_{z,\textrm{los}}$
$\displaystyle=$ $\displaystyle-\frac{V^{2}_{LSR}}{R_{\odot}}\left(\cos
l+\frac{\beta}{\beta^{2}+\sin^{2}l}\right)\cos b$ (16)
$\displaystyle-\frac{2.27|z_{\textrm{kpc}}|+3.68(1-e^{-4.31|z_{\textrm{kpc}}|})}{10^{-9}\textrm{
cm s}^{-2}}|\sin b|,$

where $l$ and $b$ are the Galactic longitude and latitude coordinates of the
source, $\beta=(d/R_{\odot})\cos b-\cos l$, where $d$ is the distance of the
source from the Sun, and $z_{\textrm{kpc}}$ is the height from the midplane in
kpc. The trigonometry of the model is not relevant to the Galactic potential,
rather it provides the projection of the Galactic acceleration along our line-
of-sight based on the source’s position.

As this potential is separable, we can write the vertical acceleration of this
model as

$a_{z}=-A|z|-B\left(1-e^{-C|z|}\right),$ (17)

which corresponds to a potential of

$\Phi(z)=\frac{A}{2}z^{2}+B\left(|z|+\frac{e^{-C|z|}}{C}\right),$ (18)

and in the limit of small $z$ we recover the harmonic vertical potential

$\Phi(z)\approx\frac{1}{2}(A+BC)z^{2}=\frac{1}{2}\alpha z^{2}.$ (19)

Plugging in the values for $A$, $B$, and $C$ from Equation IV.1.4, we can
infer a value of $\log_{10}(\alpha)=3.77$ for this potential, which is
somewhat larger than our fit values of $\alpha$ for the $\alpha$,
$\alpha-\beta$, and cross potentials.

Evaluating this model for our acceleration data produces very large AIC and
$\chi^{2}$ values, indicating that the model is not a good fit to the observed
data. This is worrisome, as this model is commonly used in tests of general
relativity to obtain $\dot{P}^{\textrm{Gal}}_{b}$; if the model is not
accurate, this would lead to inconsistencies in the reported values of
$\dot{P}^{\textrm{GR}}_{b}$ in these studies. The inconsistency between the
Damour-Taylor model and the observed data appears to be due to the Damour-
Taylor model incorrectly estimating the MW disk density.

A $\Delta$AIC of 2 is positive evidence in favor of the model with the lower
AIC, while $\Delta$AIC of 6 provides strong evidence [70]. Thus, the $\alpha$,
$\alpha-\beta$, and $\alpha-\gamma$ models carry similar statistical
confidence, while MWPotential2014 [76] and MilkyWayPotential2022 [60] have
lower confidence, with the Damour-Taylor model having significantly larger AIC
and $\chi^{2}$ values relative to the other kinematic models. The extremely
large AIC of the Damour-Taylor model indicates that it should not be used to
approximate the Galactic acceleration for pulsars near the Sun.

Global Model | Parameters | AIC | $\chi^{2}$
---|---|---|---
Hernquist | $\log_{10}(M_{\textrm{tot}}/M_{\odot})=12.7\pm 0.2$ | 78 | 3.1
|  $r_{s}=32\pm 11$ kpc |  |
NFW | $\log_{10}(M_{\textrm{vir}}/M_{\odot})=13.0\pm 0.2$ | 78 | 3.1
|  $r_{s}=20\pm 7$ kpc |  |
NFW & Miyamoto-Nagai Disk | $\log_{10}(M_{\textrm{vir}}/M_{\odot})=12.8\pm 0.3$ | 84 | 3.5
|  $r_{s}=16\pm 9$ kpc |  |
| $\log_{10}(M_{\textrm{disk}}/M_{\odot})=10\pm 2$ |  |
|  $a=10\pm 40$ kpc |  |
|  $b=0.01\pm 0.5$ kpc |  |
MWPotential2014 |  Galpy Parameters [67] | 223 | 11.0
MilkyWayPotential2022 |  Gala Parameters [60] | 229 | 12.4
Table 3: Optimized fits to the global potential models.

###  IV.2 Global Models

The global models, described in Table 3, are combinations of a Hernquist
profile [77], an NFW halo [78], and a Miyamoto-Nagai Disk to fit the observed
acceleration data. These fits used all 26 sources. Note that the AIC of these
fits cannot be directly compared to the AIC of the local model fits because
they use a different set of data.

The Hernquist and NFW model fits are very similar, with total MW masses of
5-8$\times 10^{12}$ M⊙ and scale lengths in the low tens of kpc. These fits
are similar to modern potential models for the MW, except that our model fits
place the total MW mass 2-5 times larger than kinematic and dynamical models,
which place the total MW mass between 0.7 and $\sim$3$\times 10^{12}$ M⊙ [79,
80, 81, 33, 82, 83, 84, 85]. These fits are also an improvement over those of
C21, who were not able to constrain a scale radius or mass for their global
fits due to the restricted range of available data in that earlier work.

We also tested the addition of a Miyamoto-Nagai disk to an NFW potential, with
mixed results. We were able to constrain the mass of the disk (although it is
about an order of magnitude below kinematic estimates, e.g. Bovy and Rix 86,
Licquia and Newman 87, and dynamical models, e.g. Newberg  _et al._ 80), but
the scale radii of the disk potential are not well constrained.

The `Gala` MilkyWayPotential2022 model has a very poor fit to the acceleration
data compared to our optimized potential models. It should be noted, however,
that the majority of this disagreement comes from PSR B1913+16; if B1913+16 is
excluded, MilkyWayPotential2022 has AIC = 79 and $\chi^{2}=3.8$ (although the
AICs of the other models also drop by $\sim$10 when B1913+16 is removed). The
relatively poor fit of MilkyWayPotential2022 to the observed acceleration data
is in part due to the total mass estimate of the MW, which for
MilkyWayPotential2022 is about 1.1$\times 10^{12}$ M⊙.

As the Milky Way is believed to be roughly spherically symmetric over large
scales, our pulsar acceleration data should be much more sensitive to the
amount of mass within the Solar circle than outside it (via the shell
theorem). The mass enclosed within 8 kpc of the Galactic center in the NFW
global model is $M_{\textrm{enc}}(8\textrm{ kpc})=2.3\times 10^{11}$ M⊙, which
is $\sim$2.3 times larger than the enclosed mass of the MilkyWayPotential2022
model ($M_{\textrm{enc}}(8\textrm{ kpc})=9.8\times 10^{10}$ M⊙). Thus, the
pulsar acceleration data imply a larger MW mass than kinematic methods. If
these larger masses are to be believed, they imply substantial differences
between orbits calculated using potentials calibrated to modern kinematic data
and the potentials fit to the pulsar data. However, it should be noted that
the current global model fits estimate an extremely large circular speed at
the location of the Sun (roughly 320 km/s), so it is likely that we are not
yet producing precise constraints on the enclosed mass, and it is expected
that the addition of future pulsar data will significantly improve these fits.

##  V Inferring Disequilibrium From Accelerations

One way that accelerations can be used to probe disequilibrium in the MW disk
is by looking at the vertical acceleration profile ($a_{z}$ vs. $z$).
Chakrabarti  _et al._ 25 showed that simulations of a MW interacting with
dwarf galaxies would have a vertical acceleration profile that is highly
asymmetric about the midplane. As a result, showing that
$a_{z}(+z^{\prime})\neq a_{z}(-z^{\prime})$ for some nonzero $z^{\prime}$, or
alternatively that $\derivative*{a_{z}}{z}$ in the midplane is nonzero,
indicates disequilibrium in the Galactic disk.

Figure 5: Observed and simulated asymmetry in the vertical acceleration profile of the disk. The fit of the anharmonic potential profile to the observed pulsar data is shown as the black line, and 1$\sigma$ uncertainties are shown as the shaded gray region. The vertical acceleration profile from a simulation of the MW and the Sgr dSph [18] is shown as a blue line with error bars. The shape of the two profiles are roughly the same, suggesting that the observed acceleration asymmetry is consistent with the effects of orbiting dwarf galaxy interactions. Model | A | B | $\derivative*{V_{\textrm{circ}}}{R}$ | $\rho_{0}$ | $\rho_{\textrm{0,DM}}$
---|---|---|---|---|---
| (km/s/kpc) | (km/s/kpc) | (km/s/kpc) | (M⊙/pc3) | (M⊙/pc3)
$\alpha-\beta$ | 12.8$\pm$2.4 | -15.7$\pm$2.4 | 4$\pm$5 | 0.060$\pm$0.018 | -0.012$\pm$0.018
$2\alpha-\beta$ | 15.4$\pm$2.6 | -13.1$\pm$2.6 | -2$\pm$5 | 0.062$\pm$0.017 | -0.010$\pm$0.018
$\alpha-\beta$ \+ 2PM | 20.4$\pm$1.7 | -8.1$\pm$1.7 | -12$\pm$5 | 0.066$\pm$0.017 | -0.006$\pm$0.018
Table 4: Oort constants, the slope of the rotation curve, and the (dark
matter) density in the midplane for different potential models.

###  V.1 Asymmetric Local Models

We now move on to models that capture disequilibrium features of the Galactic
potential, i.e. these potentials are not invariant under negation of $\phi$
and/or $z$.

In general, the more flexible asymmetric models do a better job of fitting the
observed acceleration data. This is strong evidence that the binary pulsar
acceleration data contains substantial disequilibrium effects, which must be
accounted for in order to obtain an accurate model of the Galactic potential.

####  V.1.1 Anharmonic and 2$\alpha$-$\beta$ Models

The first disequilibrium model is the anharmonic model, which consists of the
$\alpha$ model except the vertical component of the potential is expanded one
degree further in a power series expansion for a separable potential:

$\Phi(z)=\sum_{n=2}^{3}\frac{\alpha_{n-1}}{n}z^{n},$ (20)

which provides a simple form for $\derivative*{a_{z}}{z}$ in the midplane:

$\evaluated{\derivative{a_{z}}{z}}_{z=0}=\evaluated{\derivative{z}\left(-\derivative{\Phi}{z}\right)}_{z=0}=-\alpha_{1}.$
(21)

In essence, $\alpha_{1}$ can be interpreted as a measure of how much the local
disk potential is out of equilibrium.

The additional $z^{3}$ term allows us to characterize the asymmetry in the
vertical acceleration profile, which is shown in Figure 5. The observed value
of $a_{z}$ below the disk has a smaller magnitude than $a_{Z}$ above the disk,
which is consistent with simulations of the MW interacting with the Sgr dSph.
The methodology of the simulations is described in [16, 17], and are the same
simulations that are used in [18]. This is consistent with the popular idea
that this type of disequilibrium is (at least in part) caused by the disk
interacting with orbiting satellites.

The $2\alpha-\beta$ model is identical to the anharmonic model except that it
does not require a flat rotation curve; the interpretation of the $\beta$
parameter is the same as for the $\alpha-\beta$ model.

We also attempted to fit a model that included a $z^{4}$ term, although this
quartic term could not be constrained with existing data. A model that
includes a quartic term would be useful for quantifying the behavior of the
phase space spiral that has been observed in the MW disk [4].

####  V.1.2 Local Expansion Model

The local expansion model is a 1st-order Taylor series expansion of the
acceleration profile centered at the position of the Sun assuming that the
potential is additively separable:

$a_{i}(\mathbf{x})\approx
a_{i}(\mathbf{x}_{\odot})+{\partialderivative{a_{i}}{x_{i}}}\bigg{\rvert}_{\odot}(x_{i}-x_{i,\odot}).$
(22)

While it is useful to characterize the slope of the rotation curve with
$\derivative*{a_{R}}{R}$ and the midplane density of the disk with
$\derivative*{a_{z}}{z}$, these terms do not describe disequilibrium features.
The nonzero $\derivative*{a_{\phi}}{\phi}$ implies disequilibrium, as it
requires that the potential be a function of azimuth, rather than
axisymmetric. The AIC of this model is a substantial improvement over the
anharmonic and $2\alpha-\beta$ models, which indicates that the MW disk
density has substantial azimuthal variation in the Solar neighborhood.

####  V.1.3 Sinusoidal Model

The MW disk has been shown to contain ripples, which are probably related to
the passage of orbiting satellites [3, 88, 20, 89]. We explore these features
with the sinusoidal model, which follows the same idea as the density model
outlined in Xu  _et al._ 3 of varying the “midplane” of the potential as a
sine curve, except we use a harmonic oscillator disk potential instead of an
exponential density profile. This model outperforms the anharmonic and
$2\alpha-\beta$ models, suggesting that the disk ripples are a prominent
feature of the disk potential. We obtain a slightly smaller wavelength and a
somewhat larger amplitude for the disk oscillations than was found in the
stellar density distribution by Xu  _et al._ 3, although these differences
could be largely due to the different definitions of the two models; we used a
conservative model in order to minimize the number of free parameters that we
had to fit, whereas Xu  _et al._ 3 were able to separately fit the disk
density variations above and below the midplane.

####  V.1.4 $\alpha$-$\beta$ \+ 2 Point Mass Model

The final model we discuss here is the most flexible model, which consists of
the $\alpha-\beta$ model with the addition of an arbitrary number of point
masses. The location and mass of each point mass is independently fit to the
data simultaneously with the $\alpha$ and $\beta$ parameters. This allows for
a flexible non-parametric fit to any various disequilibrium features that may
be present in the acceleration data. In order to determine the optimal number
of point masses that should be included in the model, we sequentially fit the
$\alpha-\beta$ model with $n$ point masses, increasing $n$ until the AIC no
longer improved when we added another point mass to the model. The optimal
number of point masses (which minimizes AIC) is 2. While the AIC of this model
is substantially lower than the other models in this work, the fact that its
$\chi^{2}<1$ could indicate that it is actually overfitting the data.

It is not immediately clear what the physical significance is for the point
masses in this model. On the surface, the point masses are simply a flexible
model that allows for the generation of a wide range of potential models. It
is plausible that this model is producing an approximation of a more complex
potential model, and that this presently unknown more complex model is not
well represented using our other analytical models.

One possible interpretation for these point masses is that each point mass
corresponds to a large dark matter subhalo in the Solar neighborhood. These
dark matter subhalos would have masses of roughly $10^{8}$ M⊙, which is large
but not unreasonable for a dark matter subhalo in a MW-mass galaxy [e.g. 90].
If this model is actually identifying real dark matter subhalos, then the
number density for $M\sim 10^{10}$ M⊙ subhalos must be higher than in
$\Lambda$-CDM, which predicts that there should only be a few dark matter
subhalos with this mass across the entire Galaxy [91, 92, 93]. We plan to
explore the sensitivity of direct acceleration measurements to dark matter
subhalos with a future publication, as it is outside the scope of this work.

##  VI Fundamental Galactic Parameters from Pulsar Timing

###  VI.1 Oort Constants

Characterization of the Galaxy has historically been done using the kinematics
of stars near the Sun [23]. One way to do this is by empirically estimating
the Oort constants [94], called $A$, a measure of azimuthal shear motion in
the Solar neighborhood, and $B$, a measure of the rotation curve vorticity,
i.e. the scale of epicyclic motion. These constants are related to the
rotation curve near the Sun:

$\displaystyle A-B=$
$\displaystyle\left[\frac{V_{\textrm{circ}}}{R}\right]_{R_{\odot}},\textrm{
and}$ (23) $\displaystyle A+B=$
$\displaystyle-\left[\derivative{V_{\textrm{circ}}}{R}\right]_{R_{\odot}}.$
(24)

Modern kinematic estimates from Gaia data place the Oort constants around
$A=15.1$ to $15.3$ km/s/kpc, and $B=-13.4$ to $-11.9$ km/s/kpc [76, 95], which
are consistent with the most recent estimates from Gaia DR3 [96].

From the definition of $\beta$ in the Equation 14, we can relate the
$\alpha-\beta$ model to the Oort constants:

$\beta=-\frac{A+B}{A-B},$ (25)

which implies a value of $\beta\approx-0.1$ from the Gaia values for the Oort
constants. We can also write the Oort constants as

$\displaystyle A=$
$\displaystyle\frac{1}{2}\frac{V_{LSR}}{R_{\odot}}\left(1-\beta\right),\textrm{
and}$ (26) $\displaystyle B=$
$\displaystyle-\frac{1}{2}\frac{V_{LSR}}{R_{\odot}}\left(1+\beta\right).$ (27)

The values of the Oort constants for the three $\alpha-\beta$ models in this
work are listed in Table 4. It is clear that the different models produce
substantially different values for the Oort constants, presumably because they
are simultaneously fitting different amounts of local disequilibrium. The
$2\alpha-\beta$ model is consistent with the kinematic estimates of the Oort
constants, while the other two models are inconsistent with the kinematic
estimates. These results suggest that it is not only important to consider
disequilibrium in the acceleration models in order to properly extract
properties of the Galaxy, but that it is also important how this
disequilibrium is accounted for.

Figure 6: Selected measurements of the Oort constants $A$ and $B$ since 2010.
The shape of each point indicates the type of data used to make the
measurement, and the color of each point indicates the corresponding slope of
the rotation curve given by $\derivative*{V_{\textrm{circ}}}{R}=-(A+B)$. The
values we infer from the pulsar timing data are shown as stars; while the
three models span a relatively large area on the figure, the $2\alpha-\beta$
model (the pale red star) is in broad agreement with many of the previous
measurements. The sources for this data can be found in Table 5.

Figure 6 shows our inferred values of the Oort constants compared with a
selection of measurements of the Oort constants since 2010. Our values span a
relatively large range, but the values we obtain for the Oort constants from
the 2$\alpha-\beta$ model agree well with many previous measurements.

###  VI.2 The Slope of the Rotation Curve

Equation 14 provides a way to compute the slope of the rotation curve for our
$\alpha$-$\beta$ models; the slopes of the rotation curve for each model are
provided in Table 4.

While we obtain a range of $\derivative*{V_{\textrm{circ}}}{R}$ values for our
different models, the preferred 2$\alpha$-$\beta$ model constrains the slope
of the rotation curve to be -2 $\pm$ 5 km/s/kpc at the location of the Sun.
This value is consistent with the slope of the rotation curve being flat.
However, a slightly declining rotation curve at the Solar location is in good
agreement with Gaia data, such as Figure 13 of Gaia Collaboration  _et al._
97. Additionally, Põder  _et al._ 98 contains a compilation of a few different
rotation curve models, all but one of which appear to have a value of
$\derivative*{V_{\textrm{circ}}}{R}\sim 2$ km/s/kpc at the Solar position.

###  VI.3 The Oort Limit and Local Dark Matter Density

Figure 7: The effect of different choices of scale length on the GPR map. The
choice of scale length increases from left to right. The left-most panel has
too small of a scale length, as the GPR map only estimates $a_{\textrm{los}}$
near a source and does not inteprolate between sources. The right-most panel
has too large a scale length, as it averages over complex substructure and
extrapolates $a_{\textrm{los}}$ far beyond where data is located. The middle
two panels show that for a reasonable choice of scale length, the general
shapes of the GPR maps are similar, and therefore the exact choice of scale
length is not particularly important.  Figure 8: GPR map of the observed line-
of-sight acceleration compared to the theoretical line-of-sight accelerations
from Gala MilkyWayPotential2022. The white points indicate the positions of
the binary pulsars. Red (blue) indicates an apparent line-of-sight
acceleration towards (away from) the Sun. The top row shows the $R-Z$ plane at
$Y=0$, and the bottom row shows the $X-Y$ plane at $Z=0$. The observed line-
of-sight accelerations appear to be very different from the theoretical
acceleration maps, implying a substantial departure from an equilibrium
acceleration field.  Figure 9: Tomography of the observed line-of-sight
acceleration GPR map. Colors are the same as in Figure 8, except the predicted
accelerations have been divided by the corresponding uncertainty in the
predicted GPR map in order to emphasize significance of the GPR map values.
The top row shows slices of the $R-Z$ plane at different azimuth, where $\phi$
increases in the same direction as the Sun’s rotation. The bottom row shows
slices of the $X-Y$ plane at different heights below/above the midplane. An
equilibrium acceleration field should look identical at $\pm\phi$ and $\pm Z$,
but the observed accelerations are substantially different on either side of
the Sun.

The Oort limit, or the volumetric mass density in the midplane of the disk, is
another characterization of the local Galaxy that is often derived from
kinematic data. It can be obtained by solving the Poisson Equation in
cylindrical coordinates for the $\alpha-\beta$ models, which leads to

$4\pi G\rho_{0}=\alpha+2\beta\Omega_{\odot}^{2},$ (28)

where $\Omega=V_{\textrm{circ}}/R$ is the angular velocity of the disk at a
given radius. Here, $\rho_{0}=\rho_{\textrm{bary}}+\rho_{\textrm{DM}}$ is the
midplane density of the baryonic matter plus the dark matter.

C21 used this prescription to calculate an Oort limit of
$\rho_{0}=0.08^{+0.05}_{-0.02}$ M⊙/pc3. When combined with the baryon budget
of $\rho_{0,\textrm{bary}}=0.084\pm 0.012$ M⊙/pc3 from McKee  _et al._ 99,
this results in a midplane dark matter density of
$\rho_{0,\textrm{DM}}=-0.004^{+0.05}_{-0.02}$ M⊙/pc3. C21 state that this
suggests that values of the Oort limit from Jeans analysis may be an
overestimate; their value of $\rho_{0,\textrm{DM}}$ is consistent with there
being no dark matter in the midplane, as well as McKee  _et al._ 99’s value of
$\rho_{0,\textrm{DM}}=0.013\pm 0.003~{}\rm M_{\odot}/\rm pc^{3}$.

Our results for the Oort limit and the dark matter density in the midplane are
given in Table 4. These values use a baryon budget built from a literature
compilation of the local stellar volume mass density of $\rho_{*}=0.0468\pm
0.0050$ M⊙/pc3 [100] and the gas density from McKee  _et al._ 99,
$\rho_{\textrm{gas}}=0.025\pm 0.003$ M⊙/pc3, which gives a local baryon
density of $\rho_{\textrm{bary}}=0.072\pm 0.006$ M⊙/pc3. This is similar to,
but slightly less than the baryon budget of Bienaymé  _et al._ 101:
$\rho_{0,\textrm{bary}}=0.077\pm 0.007$ M⊙/pc3.

These updated values are consistent with, and have smaller uncertainty than
the measurement of the dark matter density in the midplane from C21. These
values are all consistent with 0 dark matter in the midplane. Our measurements
of the local dark matter density are lower than recent estimates of the dark
matter density obtained from Jeans modeling (between 0.010 and 0.016
M⊙/pc3[101, 99, 102, 100, 103]), which is consistent with the claim of C21
that these kinematic methods may be overestimating the local dark matter
density. Moran  _et al._ 51 also use pulsar timing to obtain an Oort limit of
$\rho_{0}=0.036\pm 0.021$ M⊙/pc3, which is substantially lower than our
measurement by about a factor of 2, and does not appear to be consistent with
any available baryon budgets.

Note that if we instead use the McKee  _et al._ 99 baryon budget, which has a
much larger stellar mass density than the values collated in Guo  _et al._
100, we get $\rho_{0,\textrm{DM}}=-0.024\pm 0.021$, $-0.022\pm 0.021$, and
$-0.018\pm 0.021$ M⊙/pc3 for our respective models. The McKee  _et al._ 99
baryon budget is similar to recent estimates from Gaia DR2 data,
$\rho_{0,\textrm{bary}}\sim 0.09$ M⊙/pc3 [104]. Either of these baryon budgets
produce negative values for $\rho_{0,\textrm{DM}}$ that are larger than the
error bars in our models, which suggests that these baryon budgets may
overestimate the baryon content in the midplane.

Lim  _et al._ 105 used unsupervised machine learning in the form of
normalizing flows to model the phase space distribution of Gaia DR3 data. They
obtain a value of $\rho_{0}$ = 0.0617$\pm$ 0.0020 M⊙/pc3 for the Oort limit,
which is nearly identical to our value. Lim  _et al._ 105 calculate a baryon
budget of only 0.0534 $\pm$ 0.0042 M⊙/pc3 from the Gaia DR3 data; if we
instead use this baryon budget, we obtain a dark matter density in the
midplane of $\rho_{0,\textrm{DM}}$ = 0.008 $\pm$ 0.02 M⊙/pc3 for the
$2\alpha$-$\beta$ model. This value is still consistent with no dark matter in
the Galactic midplane; however, it is non-negative, which further implies that
a baryon budgets of the other works could be overestimated.

Some of these differences may be due to differences in the effective ranges
that are probed by the different studies. The “local” pulsar dataset lies
within $\sim$1 kpc of the MW disk plane, but, for example, the McKee  _et al._
99 data is primarily located within a few hundred pc of the midplane. The
differences in the vertical extents of these studies may result in somewhat
different Oort limits (since we are averaging over different vertical
extents), which means that caution must be used when directly comparing these
values.

##  VII Beyond Smooth Models

Analytical and parametric models are useful for studying Galactic structure,
because they enable one to calculate and compare fundamental Galactic
parameters from different datasets. In the regime of a static, equilibrium
Galaxy, these parametric models effectively constrain the behavior of the
Galaxy. However, we are now aware that the MW is experiencing many
disequilibrium effects, which indicates that parametric models are
approximations of the Galaxy’s behavior. These disequilibrium effects are
difficult to quantify analytically, and lend themselves to non-parametric
methods with a high degree of flexibility. Here, we develop a non-parametric
map of the acceleration field of the Galaxy, and discuss ways to improve this
map in the future.

###  VII.1 Gaussian Process Regression Maps

Gaussian Process Regression (GPR) is a non-parametric supervised learning
algorithm that trains on measurements of data, and produces an interpolated
model of the data and its uncertainties. This interpolated model can be
evaluated at any point; in our case, this allows us to build a 3-dimensional
map of the line-of-sight acceleration field within several kpc of the Sun.
This was done using the `GaussianProcessRegressor` class from `scikit-learn`
to generate the GPR model.

GPR requires a characteristic length scale for the interpolated output, which
is analogous to the scale of the kernel in a kernel density estimation.
Typically, in GPR this scale length is treated as a hyperparameter, which is
then optimized over the data in order to produce a GPR model that minimizes
the ordinary-least-squares difference between the model and the data, along
with a penalty for model complexity, similar to AIC [106]. However, the
ability of the model to successfully optimize the length-scale hyperparameter
is dependent on the number of data points available. Ideally, one would have
several dozen datapoints per dimension that could be used to obtain a good
estimate of the length scale hyperparameter [106], but this is not true in our
case. By adding or removing a single datapoint, we would obtain dramatically
different values for the length scale of the GPR model.

In order to avoid this problem, we simply set the length scale as 0.5 kpc
rather than allow the GPR method to optimize it over the dataset. The
rationale for this choice of length scale is shown in Figure 7; for too-small
choices of the length scale, the GPR model does not interpolate between
datapoints, and for too-large choices of the length scale, the GPR model
interpolates much further than is reasonable. However, for reasonable choices
of length scale (larger than the distance between datapoints but smaller than
the overall extent of the data), the GPR maps generally look the same
regardless of the exact choice of length scale.

The $R-z$ and $x-y$ planes (2-dimensional slices) from the GPR map are shown
in Figure 8, along with the line-of-sight accelerations that are expected to
be observed based on the MilkyWayPotential2022 potential model. It is
immediately clear that the general shapes of the GPR map and the theoretical
acceleration maps do not match, which implies that the observed acceleration
data is not well-modeled by an equilibrium acceleration profile of the Galaxy.
However, note that the theoretical $a_{\textrm{los}}$ model has structure;
this is a map of the tidal field, which is squeezed in the direction of
rotation and elongated along the direction towards the Galactic center.

A useful benchmark for the scale of the observed acceleration perturbations is
to consider the radial and vertical components of the acceleration at the
location of the Sun. The radial acceleration is roughly

$a_{R}\sim\frac{V_{\textrm{circ}}^{2}}{R_{\odot}}=6.8\textrm{ mm/s/yr}.$ (29)

Assuming an infinite density sheet with a surface density of $\Sigma\sim$50
M⊙/pc2, the vertical acceleration is

$a_{z}\sim G\Sigma=0.2\textrm{ mm/s/yr}.$ (30)

The observed variations are on the scale of 0.5-1 mm/s/yr, which is sizeable
compared to the vertical acceleration but a fraction of the radial
acceleration scale. We are only able to observe variations in
$a_{\textrm{los}}$ , which is a projected combination of $a_{R}$ and $a_{z}$,
so the variations in $a_{\textrm{los}}$ we see are on the $\sim 10\%$ scale of
the total acceleration.

Further, we provide the tomography of the GPR map in Figure 9. If the Galaxy
were in equilibrium, the slices of the map at positive and negative $\phi$
and/or $z$ would be identical, because the Galaxy would be azimuthally
symmetric. However, it is clear from the tomography that this is not the case;
for example, the slice at $\phi=-5^{\circ}$ looks substantially different than
the slice at $\phi=+5^{\circ}$. This could indicate a large amount of
disequilibrium in the Galaxy’s acceleration field near the Sun; however,
because all the acceleration points are within 1.5$\sigma$ of the model, it’s
not clear exactly how significant these variations are, or how much of the
variations are caused by the large uncertainties on the individual data
points.

This GPR map only considers the propagated uncertainty in $a_{\textrm{los}}$
for each datapoint. However, each source has its own uncertainty in distance,
which will not only change the source’s position on the map, but the distance
uncertainty is correlated to the uncertainty in $a_{\textrm{los}}$ in a non-
linear way. In the future we plan on creating GPR maps that take this type of
correlated uncertainty in the input and output of each datapoint into account.
Despite these shortcomings, the proof-of-concept GPR maps presented here are
still powerful diagnostics for disequilibrium features and asymmetries in the
acceleration profile of the Galaxy.

##  VIII Conclusions

Here we summarize the main findings of this work:

  * •

We gather direct acceleration measurements for 26 binary pulsars, which span
3.4 kpc in $R$, and 3.6 kpc in $Z$.

  * •

The residual of the pulsar accelerations and accelerations from modern
kinematic models appear to have a global gradient in $R$; this trend could not
be resolved by varying the mass of the theoretical potential model. However,
the trend can be eliminated if the Sun is experiencing a 1.1 mm/s/yr
acceleration away from the Galactic center, comparable to a Jupiter-mass
object at a distance of 400 AU.

  * •

We fit a collection of symmetric and asymmetric potential models to the
observed acceleration data. The more flexible asymmetric models that allow for
disequilibrium do a better job of modeling the data compared to the symmetric
potentials, indicating that the observed accelerations are consistent with the
Galactic disk being in disequilibrium. These asymmetries are consistent with
those caused by dwarf galaxies interacting with the MW disk, although it is
difficult to determine the exact physical mechanism responsible for the
asymmetries given the current limitations of the acceleration data.

  * •

The Damour-Taylor Potential [29], which is commonly used to obtain an
approximation of the Galactic acceleration in tests of general relativity, is
not a good fit to the observed acceleration data. Studies that use this
potential to calibrate the observed acceleration of sources due to the
gravitational field of the MW will therefore produce systematically inaccurate
results, especially for sources with large heights from the Galactic midplane.

  * •

We constrain fundamental Galactic parameters using the acceleration data. Our
values of the Oort Constants vary substantially depending on which model we
use, but a disequilibrium model that considers the asymmetry of the vertical
acceleration profile produces values for the Oort constants of $A=15.4\pm 2.6$
km/s/kpc and $B=-13.1\pm 2.6$ km/s/kpc, which are consistent with literature
values from kinematic estimates. This model also produces a 3.6$\sigma$
measurement of the Oort limit, or the density in the midplane, of
$\rho_{0}=0.062\pm 0.017$ M⊙/pc3, and a dark matter density in the midplane of
$\rho_{0,\textrm{DM}}=-0.010\pm 0.018$ M⊙/pc3. However, if we use the baryon
budget of Lim  _et al._ 105, we obtain a positive dark matter density in the
midplane of $\rho_{0,\textrm{DM}}=0.008\pm 0.02$ M⊙/pc3. The inferred dark
matter density is consistent with there being no dark matter density in the
Galactic plane.

  * •

We constrain the slope of the rotation curve to be -2 $\pm$ 5 km/s/kpc at the
location of the Sun. This is consistent with the slope of the rotation curve
being flat, but it is also consistent with the Gaia findings that the
azimuthal velocity curve in the plane is slightly decreasing at the Solar
location.

  * •

We measure the oblateness of the potential using a potential model that
includes a cross ($\gamma$) term. Our measured value of $\gamma$ relies on
updating pulsar timing solutions and produces a similar oblateness and scale
height to thick disk kinematic data from Sharma  _et al._ 72, in contrast to
the C21 dataset, which indicated a more oblate potential near the Galactic
midplane.

  * •

We provide non-parametric maps of the acceleration field of the Galaxy, which
exhibit a substantial amount of variation from the theoretical acceleration
maps. The tomography of this map is consistent with a substantial amount of
disk disequilibrium, given its asymmetry in $Z$ and $\phi$.

As binary pulsar datasets continue to improve and grow, we will be able to
produce progressively better constraints on the MW’s gravitational potential
and its dark matter content. While more work remains to determine the sources
of the currently unexplained variations in accelerations compared to
theoretical maps, we have now illustrated the need for flexible disequilibrium
models in order to utilize the acceleration data to its full capabilities.

###### Acknowledgements.

This work utilizes data from the Australia Telescope National Facility
Catalogue [35], which can be found online at
http://www.atnf.csiro.au/research/pulsar/psrcat. MTL graciously acknowledges
support received from NSF AAG award number 2009468, and NSF Physics Frontiers
Center award number 2020265, which supports the NANOGrav project. SC
gratefully acknowledges support from NSF AST 2009828. Software used: Numpy
[107], Scipy [69], Sci-kit Learn [108], Astropy [109], Matplotlib [110], emcee
[111], Gala [60], Galpy [67, http://github.com/jobovy/galpy]

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##  Appendix A Oort Constants Data

In Table 5 we provide the data and references used to make Figure 6.

$A$ | $B$ | Reference | Comments
---|---|---|---
(km/s/kpc) | (km/s/kpc) |  |
17.8 $\pm$ 0.8 | -13.2 $\pm$ 1.5 | [112] | Masers
14.85 $\pm$ 7.47 | -10.85 $\pm$ 6.63 | [113] | Hipparcos F Giants
17.42 $\pm$ 1.17 | -12.46 $\pm$ 0.86 | [114] | Hipparcos Cepheids, 0.2 kpc $<r<$ 3.0 kpc
18.17 $\pm$ 1.14 | -11.91 $\pm$ 0.85 |  | Above, 0.5 kpc $<r<$ 3.0 kpc
17.9 $\pm$ 0.5 | -13.6 $\pm$ 1.0 | [115] | OB III Stars
14.05 $\pm$ 3.28 | -9.30 $\pm$ 2.87 | [116] | Hipparcos G III Stars
9.2 $\pm$ 0.5 | -17.4 $\pm$ 0.5 | [117] | RAVE
16.9 $\pm$ 1.2 | -13.5 $\pm$ 1.4 | [118] | Masers
16.7 $\pm$ 0.6 | -12.0 $\pm$ 1.0 | [119] | Masers
16.00 $\pm$ 0.36 | -14.17 $\pm$ 0.28 | [120] | Hipparcos OB Stars
14.2 $\pm$ 0.2 | -10.0 $\pm$ 0.2 | [121] | UCAC4 11th Mag
9.1 $\pm$ 0.1 | -10.9 $\pm$ 0.1 |  | Above, UCAC4 16th Mag
12.1 $\pm$ 0.7 | -10.6 $\pm$ 0.5 |  | Above, PPMXL 11th Mag
10.3 $\pm$ 0.2 | -12.1 $\pm$ 0.2 |  | Above, PPMXL 17th Mag
12.2 $\pm$ 0.4 | -5.8 $\pm$ 0.3 |  | Above, XPM 11th Mag
7.0 $\pm$ 0.1 | 10.0 $\pm$ 0.1 |  | Above, XPM 17th Mag
17.12 $\pm$ 0.45 | -11.60 $\pm$ 0.77 | [122] | RAVE, $r<250$ pc
16.53 $\pm$ 0.52 | -10.82 $\pm$ 0.93 | [123] | Gaia DR1 OB Stars
17.77 $\pm$ 0.46 | -13.76 $\pm$ 0.71 |  | Above, but different sample
15.3 $\pm$ 0.4 | -11.9 $\pm$ 0.4 | [76] | Gaia DR1, $r<230$ pc
15.57 $\pm$ 0.31 | -10.72 $\pm$ 0.5 | [124] | RAVE5 $+$ TGAS
15.07 $\pm$ 0.25 | -12.17 $\pm$ 0.39 | [125] | TGAS $+$ RAVE
14.13 | -6.7 | [126] | TGAS-RAVE-LAMOST, Young Stars
18.8 | -12.7 |  | Above, Intermediate Age Stars
17 | -2.1 |  | Above, Old Stars
16.29 $\pm$ 0.06 | -11.9 $\pm$ 0.05 | [127] | TGAS MS Stars, $r<1.5$ kpc
13.32 $\pm$ 0.09 | -12.71 $\pm$ 0.06 |  | Above, TGAS RG Stars, $r<1.5$ kpc
15.1 $\pm$ 0.1 | -13.4 $\pm$ 0.1 | [95] | Gaia DR2, $r<500$ pc
15.73 $\pm$ 0.32 | -12.67 $\pm$ 0.34 | [128] | Gaia DR2
15.6 $\pm$ 1.6 | -13.9 $\pm$ 1.8 | [129] | SEGUE Halo RG Stars
16.31 $\pm$ 0.89 | -11.99 $\pm$ 0.79 | [130] | LAMOST A Stars, $r<600$ pc, $|z|<100$ pc
16.2 $\pm$ 0.2 | -11.7 $\pm$ 0.2 | [131] | Gaia DR2 “(p)”
15.2 $\pm$ 0.8 | -12.4 $\pm$ 0.9 |  | Above, “(v)”
15.6 $\pm$ 1.6 | -15.8 $\pm$ 1.7 | [132] | GCNS, Gaia EDR3
Table 5: Selected measurements of the Oort constants $A$ and $B$ since 2010.