Orbital Torus Imaging: Acceleration, Density, and Dark Matter in the Galactic Disk Measured with Element Abundance Gradients

<jats:p>Under the assumption of a simple and time-invariant gravitational potential, many Galactic dynamics techniques infer the milky Way’s mass and dark matter distributions from stellar kinematic observations. These methods typically rely on parameterized potential models of the Galaxy and must take into account nontrivial survey selection effects, because they make use of the density of stars in phase space. Large-scale spectroscopic surveys now supply information beyond kinematics in the form of precise stellar label measurements (especially element abundances). These element abundances are known to correlate with orbital actions or other dynamical invariants. Here, we use the Orbital Torus Imaging framework that uses abundance gradients in phase space to map orbits. In many cases these gradients can be measured without detailed knowledge of the selection function. We use stellar surface abundances from the Apache Point Observatory Galactic Evolution Experiment survey combined with kinematic data from the Gaia mission. Our method reveals the vertical (<jats:italic>z</jats:italic>-direction) orbital structure in the Galaxy and enables empirical measurements of the vertical acceleration field and orbital frequencies in the disk. From these measurements, we infer the total surface mass density, Σ, and midplane volume density, <jats:italic>ρ</jats:italic> <jats:sub>0</jats:sub>, as a function of Galactocentric radius and height. Around the Sun, we find <jats:inline-formula> <jats:tex-math> <? CDATA ${{\rm{\Sigma }}}_{\odot }(z=1.1\,\mathrm{kpc})={72}_{-9}^{+6}\,{M}_{\odot }\,{\mathrm{pc}}^{-2}$? > </jats:tex-math> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1.1</mml:mn> <mml:mspace width="0.25em" /> <mml:mi>kpc</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>72</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>9</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msubsup> <mml:mspace width="0.50em" /> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:mspace width="0.25em" /> <mml:msup> <mml:mrow> <mml:mi>pc</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="apjad16e8ieqn1.gif" xlink:type="simple" /> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math> <? CDATA ${\rho }_{\odot }(z=0)={0.081}_{-0.009}^{+0.015}\,{M}_{\odot }\,{\mathrm{pc}}^{-3}$? > </jats:tex-math> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>0.081</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.009</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.015</mml:mn> </mml:mrow> </mml:msubsup> <mml:mspace width="0.50em" /> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:mspace width="0.25em" /> <mml:msup> <mml:mrow> <mml:mi>pc</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="apjad16e8ieqn2.gif" xlink:type="simple" /> </jats:inline-formula> using the most constraining abundance ratio, [Mg/Fe]. This corresponds to a dark matter contribution in surface density of Σ<jats:sub>⊙,DM</jats:sub>(<jats:italic>z</jats:italic> = 1.1 kpc) = 24 ± 4 <jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub> pc<jats:sup>−2</jats:sup>, and in total volume mass density of <jats:italic>ρ</jats:italic> <jats:sub>⊙,DM</jats:sub>(<jats:italic>z</jats:italic> = 0) = 0.011 ± 0.002<jats:italic> M</jats:italic> <jats:sub>⊙</jats:sub> pc<jats:sup>−3</jats:sup>. Moreover, using these mass density values we estimate the scale length of the low-<jats:italic>α</jats:italic> disk to be <jats:italic>h</jats:italic> <jats:sub> <jats:italic>R</jats:italic> </jats:sub> = 2.24 ± 0.06 kpc. </jats:p>