We present a model for the [ \alpha / { Fe } ] - [ { Fe } / { H } ] distribution of stars in the inner Galaxy , 3 { kpc } < R < 5 { kpc } , measured as a function of vertical distance |z| from the midplane by Hayden et al .
( 2015 , H15 ) .
Motivated by an “ upside-down ” scenario for thick disk formation , in which the thickness of the star-forming gas layer contracts as the stellar mass of the disk grows , we combine one-zone chemical evolution with a simple prescription in which the scale-height of the stellar distribution drops linearly from z _ { h } = 0.8 { kpc } to z _ { h } = 0.2 { kpc } over a timescale t _ { c } , remaining constant thereafter .
We assume a linear-exponential star-formation history , \dot { M } _ { * } ( t ) \propto te ^ { - t / t _ { sf } } .
With a star-formation efficiency timescale \tau _ { * } = M _ { g } ( t ) / \dot { M } _ { * } ( t ) = 2 { Gyr } , an outflow mass-loading factor \eta = \dot { M } _ { out } ( t ) / \dot { M } _ { * } ( t ) = 1.5 , t _ { sf } = 3 { Gyr } , and t _ { c } = 2.5 { Gyr } , the model reproduces the observed locus of inner disk stars in [ \alpha / { Fe } ] - [ { Fe } / { H } ] and the metallicity distribution functions ( MDFs ) measured by H15 at |z| = 0 - 0.5 { kpc } , 0.5 - 1 { kpc } , and 1 - 2 { kpc } .
Substantial changes to model parameters lead to disagreement with the H15 data ; for example , models with t _ { c } = 1 { Gyr } or t _ { sf } = 1 { Gyr } fail to match the observed MDF at high- |z| and low- |z| , respectively .
The inferred scale-height evolution , with z _ { h } ( t ) dropping on a timescale t _ { c } \sim t _ { sf } at large lookback times , favors upside-down formation over dynamical heating of an initially thin stellar population as the primary mechanism regulating disk thickness .
The failure of our short- t _ { c } models suggests that any model in which thick disk formation is a discrete event will not reproduce the continuous dependence of the MDF on |z| found by H15 .
Our scenario for the evolution of the inner disk can be tested by future measurements of the |z| -distribution and the age-metallicity distribution at R = 3 - 5 { kpc } .