We have used the tidal equations of Zahn to determine the maximum orbital distance at which companions are brought into Roche lobe contact with their giant primary , when the primary expands during the giant phases .
This is a key step when determining the rates of interaction between giants and their companions .
Our stellar structure calculations are presented as maximum radii reached during the red and asymptotic giant branch ( RGB and AGB , respectively ) stages of evolution for masses between 0.8 and 4.0 M _ { \odot } ( Z=0.001 – 0.04 ) and compared with other models to gauge the uncertainty on radii deriving from details of these calculations .
We find overall tidal capture distances that are typically 1-4 times the maximum radial extent of the giant star , where companions are in the mass range from 1 M _ { J } to a mass slightly smaller than the mass of the primary .
We find that only companions at initial orbital separations between \sim 320 and \sim 630 R _ { \odot } will be typically captured into a Roche lobe-filling interaction or a common envelope on the AGB .
Comparing these limits with the period distribution for binaries that will make PN , we deduce that in the standard scenario where all \sim 1-8 M _ { \odot } stars make a PN , at most 2.5 per cent of all PN should have a post-common envelope central star binary , at odds with the observational lower limit of 15-20 per cent .
The observed over-abundance of post-interaction central stars of PN can not be easily explained considering the uncertainties .
We examine a range of explanations for this discrepancy .