The velocity dispersion of nearby stars in the Galactic disc are well known to increase substantially with age ; this is the so-called Age-Velocity relation , and is interpreted as a “ heating ” of the disc as a function of time .
We have studied the heating of the Galactic stellar disc due to giant molecular clouds and halo black holes , via simulations of the orbits of tracer stars embedded in a patch of the local Galactic disc .
We examine a range of masses and number densities of the giant molecular cloud and halo black hole perturbers .
The heating of the stellar disc in the simulations is fit with a simple power law of the \sigma \propto t ^ { \alpha } where \sigma is the velocity dispersion of the tracer stars as a function of time , t .
We also fit this form to the best determinations of the increase in the velocity dispersion as a function of time as derived from stars in the solar neighbourhood for which ages can be reliably assigned .
Observationally , \alpha is found to lie in the range 0.3 to 0.6 , i.e .
it remains poorly constrained and its determination is probably still dominated by systematic errors .
Better constrained observationally is the ratio of the velocity dispersions of the stars in the vertical z and horizontal x ( i.e .
toward the Galactic center ) directions , being \sigma _ { z } / \sigma _ { x } = 0.5 \pm 0.1 .

For the heating of the stellar disc due to giant molecular clouds ( GMCs ) we derive a heating \sigma \propto t ^ { 0.21 } , which differs somewhat from early ( analytic ) studies in which \sigma \propto t ^ { 0.25 } .
This confirms the well known results that there are insufficient GMCs to heat the Galactic disc appropriately .
A range of dark halo black hole scenarios are verified to heat the stellar disc as \sigma \propto t ^ { 0.5 } ( as expected from analytical studies ) , and give \sigma _ { z } / \sigma _ { x } in the range 0.5 to 0.6 , which is consistent with observations .
Black holes with a mass of 10 ^ { 7 } { M } _ { \odot } are our favoured disc heaters , although they are only marginally consistent with observations .
Simulations featuring a combination of giant molecular clouds and halo black holes can explain the observed heating of the stellar disc , but since other perturbing mechanisms , such as spiral arms , are yet to be included , we regard this solution as ad hoc .