How long (weeks, month, years) does it normally take engineers to find good interplanetary trajectories, such as Cassini, Messenger or BepiColombo?
I would be particularly interested in the time required for the MESSENGER trajectory. I found some reference on NASA's Solar System Exploration: News & Events, which states that it was a "challenge", but unfortunately no specific time.
The big question is, when do you start timing it and when do you stop? For MESSENGER (you mention you're most interested in James V. McAdams' work on its trajectory analysis, for which he was named the 2008 Engineer of the Year by the AIAA Baltimore Section), McAdams himself starts one of the related publications (MESSENGER MERCURY ORBIT TRAJECTORY DESIGN, AAS 03-209, PDF) as:
After nearly three decades of Mercury orbiter spacecraft mission studies, recent improvements in ballistic trajectory design and spacecraft technology opened the door to low-cost mission options.
And later in the same document:
During the last 15 years, space agencies around the world have invested heavily in studies of “comprehensive” Mercury orbiter missions that often require advanced propulsion, two orbiting spacecraft, and sometimes a lander or surface penetrator.
So as you can see, it's tough to precisely pinpoint when MESSENGER trajectory analysis started. So I would suggest looking for when mission specific information was made available, to enable further refinement of its final trajectory. I would suggest the date of publishing the 1998 Discovery Program Announcement of Opportunity (AO), which (quoting IMPROVEMENTS IN TRAJECTORY OPTIMIZATION FOR MESSENGER: THE FIRST MERCURY ORBITER MISSION, AAS 01-458, PDF):
... imposed restrictions on launch dates, available launch vehicles, and total mission cost. Thermal protection, propellant mass fraction, and a comprehensive science payload left one choice – the Delta 2925H-9.5 (formerly 7925H-9.5), which is the highest-performance expendable launch vehicle allowed within Discovery. The Delta 2925H-9.5 launches from Cape Canaveral Air Force Station, thereby limiting declination of launch asymptote (DLA) to ± 28.5˚ without performance degradation. The AO guideline that “a mission launch every 12 to 24 months,” the AO requirement “that launch can take place by September 30, 2004,” and a Discovery mission (CONTOUR) launch set for July 1, 2002, all constrain any MESSENGER baseline and backup launch opportunities to begin and end between July 1, 2003, and September 30, 2004.
Announcement of Opportunity: Discovery Program, NASA AO 98-OSS-04, was published on March 31, 1998.
So now all you have to do is decide when to stop considering any work being done on MESSENGER's initial trajectory analysis. Perhaps its launch date on August 3, 2004? Or maybe its first Mercury flyby in January 2008? Depends on how much of its trajectory would you like to include in this timing of yours, but MESSENGER is currently still on its extended mission since March 17, 2012. And its trajectory is still being designed and periodically corrected. Your call.
For also a pretty interesting read, here's Takuto Ishimatsu's thesis for Master of Science in Aeronautics and Astronautics at Massachusetts Institute of Technology: Interplanetary trajectory analysis for 2020-2040 Mars missions including Venus flyby opportunities (35.61 Mb PDF). And if you're interested in more convoluted trajectories, I'd suggest looking at the work on the ICE/ISEE-3 one. It's simply fantastic, but it's still being worked on too, 36 years after its launch. I currently don't have a good link for it's trajectory design available, but it shouldn't be too difficult to find with a web search, it's been all over the news recently.
So you've done all the work to pick a trajectory type, picked the optimum point to hit on the arrival, decided on a 60 day launch window,... but then you still have to tell your launch vehicle or upper stage where you prefer to be delivered at any point in that entire period. More solutions needed! Almost 30 years ago the process for that step for use with the Inertial Upper Stage (IUS) for Magellan, Galileo and Ulysses missions involved JPL creating dozens of time limited overlapping sets of desired energy and outgoing asymptote specifications in polynomial form that were loaded as data into the IUS guidance system as targets. At this point you still can't go on vacation. JPL also constructed figure-of-merit matrices that were useful to estimate the cost to the spacecraft from injection errors (basically estimating the delta-V of a maneuver 10 days after launch). These were used to make decisions in real time during the upper stage mission. It probably took them a couple of months just to do all this mission design. (Actually, I assume they did it twice. Once for development use and once as close to launch as possible... finishing at something like L-6 months.)
