A spiral galaxy can be approximated as an infinitesimally thin disc of a uniform surface mass density (mass per unit area) located at $z=0,$ Two stars $A$ and $B$ start from rest from heights $2z_0$ and $z_0$ ($z_0<<$ radial extent of the disc), respectively and fall towards the disc, cross over to the other side and execute periodic oscillations. The ratio of time periods of $A$ and $B$ is

Spiral galaxies consists of flat rotating disc containing stars, gas and dust.

Now, we thought of spiral galaxy in form of a disc.

Gravitational intensity at distance $r$$r$ over axis is

$I_g=\frac{2GM}{a^2}\left(1-\frac{r}{{\left(r^2+a^2\right)}^{1/2}}\right)$

$=\frac{2GM}{a^2}\left(1-\frac{1}{{\left(1+\frac{a^2}{r^2}\right)}^{1/2}}\right)$

$=\frac{2GM}{a^2}\left(1-{\left(1+\frac{a^2}{r^2}\right)}^{-\frac{1}{2}}\right)$

Using binomial approximation, we have
$I_g\approx \frac{2GM}{a^2}\left(1-1+\frac{1}{2}\frac{a^2}{r^2}\right)\approx \frac{2GM}{r^2}$

So, forces on planet $A$ and $B$ are
$F_A=\frac{2GM{m}_A}{4z_0^2},F_B=\frac{2GM{m}_B}{z_0^2}$

Accelerations of $A$ and $B$ respectively, are
$a_A=\frac{GM}{2z_0^2}$ and $a_B=\frac{2GM}{z_0^2}$

If $T_A$ and $T_B$ are time periods of oscillations of $A$ and $B$, then

$\frac{T_A}{T_B}=\sqrt{\frac{a_B}{a_A}}$ [we use $T\propto \frac{1}{\sqrt{a}}$ as in case of a linear oscillator]

$=\sqrt{\frac{4}{1}}\Rightarrow \frac{T_A}{T_B}=2$