note: it's been a year and a half and Capstone is scheduled to be launched soon, so perhaps there is more information available now?
The Verge's How small launcher Rocket Lab plans to pull off its first mission to the Moon next year says (though this has since been addressed†):
Photon is going to need it all in order to get NASA’s CAPSTONE into the Moon’s orbit. After launching from Rocket Lab’s new launch site in Virginia, Electron will first deploy the Photon spacecraft — with the CAPSTONE satellite mounted on top — into a low circular orbit around Earth. From there, Photon will slowly stretch its orbit, taking the vehicle farther away from the planet. Eventually, Photon will burn its HyperCurie engine and set it on a course toward the Moon. All in all, it’ll take about eight to nine days to reach the right spot where Photon will deploy the CAPSTONE spacecraft, and Stiles says the route they’re taking will save Rocket Lab some energy along the way. “That’s a pretty key enabler for the mission,” says Stiles.
It's not uncommon to raise the apogee of an uncrewed spacecraft starting from LEO in several steps rather than one long burn. See for example
- Does Chandrayaan 2 use gravity assist?
- Trajectory Optimization To The Moon (Case Study: Chandrayaan-2)
- Chandrayaan-2; time taken to get to the Moon
- What was Chandrayaan-1's ground track?
- Why is the Chandrayaan lunar insertion so long?
and possibly these low-energy transfer-related questions
- Low Energy Transfer within Earth-Moon system
- Going from LEO to lunar using only low-thrust ion propulsion - can it be done?
Eight to nine days is tiny for a mission that will likely last several years, there's no food or water to run out as there would be for a crewed mission.
I'm just wondering if the eight or nine days is due to several apogee-raising burns at perigee, or if it's just one or two plus the delay associated with the natural slowing way down near the Earth-Moon L1 point on a low-energy transfer between Earth's sphere of influence and the Moon's.
Question: How many times will the Photon fire its Curie in order to "reach the right spot" to deploy CAPSTONE towards the Moon? If known, which apogee will it reach after each one?
†The block quote may be misleading. This comment states:
...NASA is paying for CAPSTONE, just as it is paying for the launch, but the spacecraft idea came from Advanced Space, it is owned by Advanced Space, all of the mission design and payload has been researched literally for years by Advanced Space, and all of the flight dynamics operations will be handled by Advanced Space