Most fuel-efficient way to navigate a rocket between two planet surface points

[mk-fg]

Imagine you've landed on a planet in one of the biomes.

What will be the most fuel-efficient way to move from there to another (specific) point on the surface?

And - more importantly - what are the important variables there?

I'm using SCAN mod which allows to have navigation satellites in orbit to provide in-game map of biomes on the surface.

And naturally, if there's some extra fuel left, I'd like to lift off and fly to another biome than the one I've landed at initially, or go visit some anomaly on the surface.

Imagining that surface is mostly flat (wrt hills and mountains, still curved though), is it always most fuel-efficient for a powered rocket to take-off at 45deg angle from horizontal direction, burn as fast as possible to get the trajectory to the target (with an overshoot adjustment for an atmosphere drag and non-instant deceleration) and then only fire engines to decelerate as late as possible?

Or would it be more efficient for planets with atmospheres to accelerate straight up to get out of thick layers of it asap, then burn towards target and travel in a thinner layers?

And with no atmosphere and low gravity, can it be more efficient to have near-horizontal burn, not to waste fuel of Y movement?

Can it be more efficient or easy to accelerate horizontally to some fixed speed and then maintain altitude via thrust towards surface?

45deg seem to be roughly the figure used for e.g. archery on earth surface, where I think air resistance is not a huge factor (i.e. arrow is massive and aerodynamic enough), but it's a rough number for just one gravity/pressure combination, for an unpowered projectile, with certain combination of aerodynamic surfaces, which is not expected to exit the atmosphere, don't need to take earth's radius into account due to short range, etc.

ICBMs otoh seem to fly very Y-heavy trajectories exiting and re-entering the atmosphere, but this might have totally different reasoning - i.e. to gain as much velocity when approaching surface as to be nearly-uninterceptable.

Can't seem to find any threads on the subject, can someone maybe point me to these, if it's been discussed a lot already?

[Oafman]

Check out this blog for analysis of this topic: http://hopsblog-hop.blogspot.fr/

[Jesrad]

Yup, the most efficient arc for a suborbital hop in vacuum is by aiming up at 45° - angular distance / 4. You can deduce the angular distance from the radius of the planet / moon (i.e. Minmus is 60 km radius, that's 377 km circumference, so hopping 100 km away is an angular distance of 30.4 degrees). Calculating the departure velocity is a tad more complicated, and you'll have to take into consideration the altitude difference between start and finish. But then, you can just accelerate until your map plots the trajectory to the right place.

Edited June 20, 2014 by Jesrad

[mk-fg]

Thanks for the tips, latest blogpost on the link explains it perfectly, indeed.

So without atmosphere, 45deg would only be optimal on a discworld or for a relatively short hops.

Should these angles change with the atmosphere, given the drag and Isp changes there?

I imagine they should, as such most-efficient-for-vacuum arcs seem to be very low-altitude, so should it be more efficient to go up and do horizontal burn outside of the thick atmosphere layer at some point?

[mk-fg]

Should these angles change with the atmosphere, given the drag and Isp changes there?

I imagine they should, as such most-efficient-for-vacuum arcs seem to be very low-altitude, so should it be more efficient to go up and do horizontal burn outside of the thick atmosphere layer at some point?

Though I guess there might not be an easy answer, given vastly different ship engine configurations - i.e. with several powerful engines and 1 NERVA, guess it should definitely be easy to take-off and use ~800 Isp of the latter for longer trajectories...

[sojourner]

Wouldn't the most fuel efficient method on a planet with atmosphere involve wings?

[mk-fg]

Wouldn't the most fuel efficient method on a planet with atmosphere involve wings?

True that