I'm calculating the probability of collision of the satellites. I'm trying to filter the set of orbits to minimize the calculations.
Which conditions may definitely say that the given 2 orbits O1 and O2 of the satellites wouldn't intersect during 7 days propagation?
For example, if the apogee of O1 is lower than the perigee of O2 or the perigee of O1 is higher than apogee of O2.
An apogee/perigee screen is a common first step in conjunction analysis screening. Remember to add a pad at least equivalent to the miss distance you're screening for + twice the expected radial position uncertainty to the apogee/pergiee filters. And use caution with this sort of filter in very low earth orbits where apogee could be changing rapidly due to drag.
So you would reject a pair of satellites as potentially conjuncting if:
apogee1 + pad < perigee2
and vice versa. Where apogee1 is the apogee of satellite 1, perigee2 is the perigee of satellite 2.
What you're doing is geometrically screening for conjunction events close enough to warrant the additional effort to calculate probability of collision. So you set a bound on how close two objects should be before you run the calculation. That bound is the miss distance. You could use the radial distance from the JSpOC Operator Spaceflight Safety Handbook (https://www.space-track.org/documents/JSpOC_Spaceflight_Safety_Handbook_For_Operators.pdf) for the orbit regimes in question. or there are techniques to back this out of the Pc calculations based on your Pc limit, object sizes, etc. Covariance Based Pre - Filters and Screening Criteria for Conjunction Analysis
Another common filter is to calculate the times when each satellite of a pair crosses the plane of the other satellite. This can be done relatively easily & quickly. Then calculate the time difference between these nodal crossing, and sort to find those closest to each other. Then iterate to find the point of closest approach (dot product of the relative position & velocity vectors changes sign). Note that this won't work for powered flight or significant drag!
