An large amount of extra fuel is therefore needed to be onboard at liftoff to continue propelling a (typically small, always unmanned) spacecraft (such as the Gravity Probe B) into a low-earth polar orbit.
That's why the second stage is so much large than the first stage. Except that it's not.
You're really good at throwing jargon around, but you don't understand a syllable of it.
There are actual rocket scientists on this forum. I might be one of them (not saying).
You're out of your league.
Not. FWIW.
Had recently that the manned Apollo missions dwelled in low-earth orbit. This means zero or near zero elevation prior to twisting upward and around the huge donut-shaped Van Allan Belt. How wrong is this description?
Please stop insulting people who ask questions, then calling for help from friends.
Did you understand what was meant by a polar orbit (e.g., for the GP-B spacecraft)?
Consider the following:
From:
https://www.aulis.com/traj_craft.htm
Due to the tilt of the Earth’s geographic axis and its geographic poles being off-centre to the geomagnetic poles by 11.5°, this places the minimum latitudes for Van Allen’s cones of escape between 58.5° and 81.5° north or south of the equator (depending on which side of the of the Earth you are ‘escaping’ from).
Van Allen’s sentiments appear verbatim in the 2005 textbook, The Physics of Space Security by David Wright et al: “…like the proton flux, the electron flux is highest near the equator and becomes negligible at a latitude of 60° north or south.”
So for the Apollo astronauts to be able to ‘avoid’ the most dangerous parts of the belt, their outbound trajectory at the bare minimum must take them to a latitude >60° above or below the equator. This is why Van Allen had proposed back in 1959 that “manned space rockets can best take off through the radiation free zone over the poles”. Which brings us to the nub of the trajectory problem.